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stringlengths
875
949
solution
stringlengths
1
7
is_unique_solution
bool
2 classes
verdict
stringclasses
3 values
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currect_board_visualization
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146
146
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int64
0
89
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at A. Then Alice places a white stone at K. Then Bob places a black stone at J. Then Alice places a white stone at L. Where should Bob play next?
G
true
blocked
Bob
...*------*------B ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..*---|--*...|..W |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
0
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at H. Then Alice places a white stone at F. Then Bob places a black stone at L. Then Alice places a white stone at G. Where should Bob play next?
B
true
blocked
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..B---|--W...|..B |./....|./....|./ |/.....|/.....|/ *------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
1
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at E. Then Alice places a white stone at B. Then Bob places a black stone at H. Then Alice places a white stone at C. Where should Bob play next?
D
true
blocked
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| W------W------*..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
2
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at G. Then Bob places a black stone at C. Then Alice places a white stone at I. Then Bob places a black stone at D. Then Alice places a white stone at L. Where should Bob play next?
B
true
blocked
Bob
...B------B------* ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--W...|..W |./....|./....|./ |/.....|/.....|/ *------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
3
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at D. Then Alice places a white stone at E. Then Bob places a black stone at I. Then Alice places a white stone at H. Where should Bob play next?
B
true
blocked
Bob
...B------W------* ../|...../|...../| ./.|..../.|..../.| *------*------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
4
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at B. Then Alice places a white stone at C. Then Bob places a black stone at L. Then Alice places a white stone at E. Where should Bob play next?
G
true
blocked
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| W------B------*..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
5
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at E. Then Alice places a white stone at B. Then Bob places a black stone at F. Then Alice places a white stone at D. Where should Bob play next?
C
true
blocked
Bob
...W------*------* ../|...../|...../| ./.|..../.|..../.| W------W------*..| |..*---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
6
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at B. Then Alice places a white stone at F. Then Bob places a black stone at J. Then Alice places a white stone at G. Where should Bob play next?
H
true
blocked
Bob
...*------*------B ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ W------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
7
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at K. Then Alice places a white stone at C. Then Bob places a black stone at L. Then Alice places a white stone at D. Where should Bob play next?
A
true
blocked
Bob
...W------W------* ../|...../|...../| ./.|..../.|..../.| *------W------*..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ *------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
8
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at G. Then Alice places a white stone at E. Then Bob places a black stone at K. Then Alice places a white stone at F. Where should Bob play next?
B
true
blocked
Bob
...*------*------* ../|...../|...../| ./.|..../.|..../.| W------*------*..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
9
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at G. Then Bob places a black stone at F. Then Alice places a white stone at H. Then Bob places a black stone at J. Then Alice places a white stone at K. Then Bob places a black stone at L. Where should Alice play next?
B
true
blocked
Alice
...*------*------B ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..W---|--W...|..B |./....|./....|./ |/.....|/.....|/ *------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
10
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at B. Then Alice places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at K. Then Bob places a black stone at J. Where should Alice play next?
I
true
blocked
Alice
...W------B------B ../|...../|...../| ./.|..../.|..../.| W------B------*..| |..*---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
11
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at C. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at K. Then Bob places a black stone at J. Where should Alice play next?
L
true
blocked
Alice
...*------B------B ../|...../|...../| ./.|..../.|..../.| W------*------*..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
12
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at B. Then Alice places a white stone at D. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at F. Where should Alice play next?
A
true
blocked
Alice
...W------W------W ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..*---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
13
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at H. Then Bob places a black stone at B. Then Alice places a white stone at I. Then Bob places a black stone at C. Then Alice places a white stone at J. Then Bob places a black stone at F. Where should Alice play next?
G
true
blocked
Alice
...*------B------W ../|...../|...../| ./.|..../.|..../.| *------B------W..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
14
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at H. Then Bob places a black stone at I. Then Alice places a white stone at K. Then Bob places a black stone at L. Where should Alice play next?
B
true
blocked
Alice
...*------*------* ../|...../|...../| ./.|..../.|..../.| *------*------B..| |..W---|--B...|..B |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
15
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at G. Then Bob places a black stone at J. Then Alice places a white stone at H. Then Bob places a black stone at K. Where should Alice play next?
F
true
blocked
Alice
...W------B------B ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..W---|--W...|..* |./....|./....|./ |/.....|/.....|/ *------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
16
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at C. Then Alice places a white stone at K. Then Bob places a black stone at D. Where should Alice play next?
B
true
blocked
Alice
...B------B------* ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
17
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at D. Then Alice places a white stone at I. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at F. Where should Alice play next?
C
true
blocked
Alice
...B------*------W ../|...../|...../| ./.|..../.|..../.| W------*------W..| |..*---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
18
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at E. Then Alice places a white stone at C. Then Bob places a black stone at G. Then Alice places a white stone at K. Then Bob places a black stone at H. Where should Alice play next?
F
true
blocked
Alice
...*------W------* ../|...../|...../| ./.|..../.|..../.| W------*------*..| |..B---|--B...|..* |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
19
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at B. Then Alice places a white stone at C. Then Bob places a black stone at I. Then Alice places a white stone at E. Then Bob places a black stone at K. Where should Alice play next?
G
true
win
Alice
...*------W------* ../|...../|...../| ./.|..../.|..../.| W------B------B..| |..*---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
20
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at D. Then Alice places a white stone at C. Then Bob places a black stone at H. Then Alice places a white stone at K. Then Bob places a black stone at I. Where should Alice play next?
L
true
win
Alice
...B------W------* ../|...../|...../| ./.|..../.|..../.| *------W------B..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
21
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at F. Then Bob places a black stone at E. Then Alice places a white stone at H. Then Bob places a black stone at K. Where should Alice play next?
B
true
win
Alice
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
22
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at C. Then Alice places a white stone at F. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at G. Where should Alice play next?
L
true
win
Alice
...*------B------W ../|...../|...../| ./.|..../.|..../.| *------W------*..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ B------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
23
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at G. Then Bob places a black stone at E. Then Alice places a white stone at I. Then Bob places a black stone at J. Then Alice places a white stone at K. Then Bob places a black stone at L. Where should Alice play next?
C
true
win
Alice
...*------*------B ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
24
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at A. Then Alice places a white stone at J. Then Bob places a black stone at H. Then Alice places a white stone at K. Then Bob places a black stone at L. Where should Alice play next?
C
true
win
Alice
...*------*------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..B---|--*...|..B |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
25
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at B. Then Alice places a white stone at E. Then Bob places a black stone at I. Then Alice places a white stone at H. Then Bob places a black stone at K. Where should Alice play next?
A
true
win
Alice
...W------*------* ../|...../|...../| ./.|..../.|..../.| *------B------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
26
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at C. Then Alice places a white stone at F. Then Bob places a black stone at D. Then Alice places a white stone at G. Then Bob places a black stone at K. Where should Alice play next?
H
true
win
Alice
...B------B------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ W------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
27
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at A. Then Alice places a white stone at I. Then Bob places a black stone at D. Then Alice places a white stone at K. Then Bob places a black stone at H. Where should Alice play next?
B
true
win
Alice
...B------*------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
28
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at C. Then Alice places a white stone at E. Then Bob places a black stone at I. Then Alice places a white stone at H. Then Bob places a black stone at J. Where should Alice play next?
D
true
win
Alice
...*------B------B ../|...../|...../| ./.|..../.|..../.| W------*------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
29
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at A. Then Alice places a white stone at G. Then Bob places a black stone at D. Then Alice places a white stone at J. Then Bob places a black stone at L. Where should Alice play next?
F
true
fork
Alice
...B------*------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
30
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at A. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at J. Then Bob places a black stone at K. Where should Alice play next?
B
true
fork
Alice
...W------*------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ *------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
31
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at E. Then Alice places a white stone at D. Then Bob places a black stone at H. Then Alice places a white stone at J. Then Bob places a black stone at L. Where should Alice play next?
B
true
fork
Alice
...W------W------W ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..B---|--*...|..B |./....|./....|./ |/.....|/.....|/ B------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
32
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at A. Then Alice places a white stone at G. Then Bob places a black stone at K. Then Alice places a white stone at J. Then Bob places a black stone at L. Where should Alice play next?
F
true
fork
Alice
...*------*------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
33
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at D. Then Alice places a white stone at G. Then Bob places a black stone at E. Then Alice places a white stone at H. Then Bob places a black stone at I. Where should Alice play next?
B
true
fork
Alice
...B------W------* ../|...../|...../| ./.|..../.|..../.| *------*------B..| |..W---|--W...|..* |./....|./....|./ |/.....|/.....|/ B------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
34
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at D. Then Alice places a white stone at G. Then Bob places a black stone at F. Then Alice places a white stone at I. Then Bob places a black stone at L. Where should Alice play next?
C
true
fork
Alice
...B------*------* ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
35
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at D. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at H. Then Bob places a black stone at L. Where should Alice play next?
B
true
fork
Alice
...B------*------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..W---|--B...|..B |./....|./....|./ |/.....|/.....|/ W------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
36
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at B. Then Alice places a white stone at K. Then Bob places a black stone at I. Where should Alice play next?
G
true
fork
Alice
...*------*------* ../|...../|...../| ./.|..../.|..../.| B------B------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
37
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at I. Then Bob places a black stone at G. Then Alice places a white stone at K. Then Bob places a black stone at H. Then Alice places a white stone at L. Then Bob places a black stone at J. Where should Alice play next?
B
true
fork
Alice
...*------*------B ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..B---|--B...|..W |./....|./....|./ |/.....|/.....|/ *------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
38
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at F. Then Alice places a white stone at D. Then Bob places a black stone at H. Then Alice places a white stone at K. Then Bob places a black stone at J. Where should Alice play next?
B
true
fork
Alice
...W------W------B ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
39
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at E. Then Alice places a white stone at F. Then Bob places a black stone at J. Then Alice places a white stone at I. Then Bob places a black stone at L. Then Alice places a white stone at K. Where should Bob play next?
B
true
blocked
Bob
...W------*------B ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ B------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
40
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at G. Then Alice places a white stone at B. Then Bob places a black stone at I. Then Alice places a white stone at C. Then Bob places a black stone at K. Then Alice places a white stone at L. Where should Bob play next?
D
true
blocked
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| W------W------B..| |..*---|--B...|..W |./....|./....|./ |/.....|/.....|/ *------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
41
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at B. Then Alice places a white stone at H. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
F
true
blocked
Bob
...*------W------W ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
42
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at I. Then Bob places a black stone at L. Then Alice places a white stone at K. Where should Bob play next?
G
true
blocked
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ W------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
43
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at D. Then Alice places a white stone at E. Then Bob places a black stone at H. Then Alice places a white stone at I. Then Bob places a black stone at K. Then Alice places a white stone at J. Where should Bob play next?
B
true
blocked
Bob
...B------W------W ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
44
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at J. Then Bob places a black stone at H. Then Alice places a white stone at L. Where should Bob play next?
G
true
blocked
Bob
...*------W------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..B---|--*...|..W |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
45
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at B. Then Alice places a white stone at G. Then Bob places a black stone at D. Then Alice places a white stone at H. Then Bob places a black stone at K. Then Alice places a white stone at L. Where should Bob play next?
E
true
blocked
Bob
...B------*------* ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..W---|--W...|..W |./....|./....|./ |/.....|/.....|/ *------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
46
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at H. Then Bob places a black stone at L. Then Alice places a white stone at J. Where should Bob play next?
B
true
blocked
Bob
...*------W------W ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..W---|--*...|..B |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
47
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at B. Then Alice places a white stone at G. Then Bob places a black stone at I. Then Alice places a white stone at H. Then Bob places a black stone at L. Then Alice places a white stone at K. Where should Bob play next?
E
true
blocked
Bob
...*------*------* ../|...../|...../| ./.|..../.|..../.| *------B------B..| |..W---|--W...|..B |./....|./....|./ |/.....|/.....|/ *------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
48
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at A. Then Alice places a white stone at E. Then Bob places a black stone at B. Then Alice places a white stone at F. Then Bob places a black stone at K. Then Alice places a white stone at L. Where should Bob play next?
C
true
blocked
Bob
...W------*------* ../|...../|...../| ./.|..../.|..../.| B------B------*..| |..*---|--*...|..W |./....|./....|./ |/.....|/.....|/ W------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
49
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at F. Then Alice places a white stone at D. Then Bob places a black stone at I. Then Alice places a white stone at G. Then Bob places a black stone at K. Then Alice places a white stone at L. Where should Bob play next?
B
true
win
Bob
...W------*------* ../|...../|...../| ./.|..../.|..../.| W------*------B..| |..*---|--W...|..W |./....|./....|./ |/.....|/.....|/ *------B------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
50
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at F. Then Alice places a white stone at C. Then Bob places a black stone at I. Then Alice places a white stone at D. Then Bob places a black stone at K. Then Alice places a white stone at H. Where should Bob play next?
B
true
win
Bob
...W------W------* ../|...../|...../| ./.|..../.|..../.| W------*------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
51
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at F. Then Bob places a black stone at D. Then Alice places a white stone at G. Then Bob places a black stone at H. Then Alice places a white stone at I. Where should Bob play next?
E
true
win
Bob
...B------W------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..B---|--W...|..* |./....|./....|./ |/.....|/.....|/ *------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
52
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at H. Then Bob places a black stone at A. Then Alice places a white stone at I. Then Bob places a black stone at C. Then Alice places a white stone at K. Then Bob places a black stone at E. Then Alice places a white stone at L. Where should Bob play next?
G
true
win
Bob
...*------B------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..W---|--*...|..W |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
53
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at B. Then Alice places a white stone at H. Then Bob places a black stone at G. Then Alice places a white stone at J. Then Bob places a black stone at I. Then Alice places a white stone at K. Where should Bob play next?
L
true
win
Bob
...*------*------W ../|...../|...../| ./.|..../.|..../.| *------B------B..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
54
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at E. Then Bob places a black stone at G. Then Alice places a white stone at F. Then Bob places a black stone at K. Then Alice places a white stone at H. Where should Bob play next?
I
true
win
Bob
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
55
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at F. Then Alice places a white stone at C. Then Bob places a black stone at J. Then Alice places a white stone at E. Then Bob places a black stone at L. Then Alice places a white stone at I. Where should Bob play next?
B
true
win
Bob
...*------W------B ../|...../|...../| ./.|..../.|..../.| W------*------W..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
56
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at G. Then Bob places a black stone at A. Then Alice places a white stone at I. Then Bob places a black stone at D. Then Alice places a white stone at K. Then Bob places a black stone at H. Then Alice places a white stone at L. Where should Bob play next?
E
true
win
Bob
...B------*------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..B---|--W...|..W |./....|./....|./ |/.....|/.....|/ *------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
57
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at B. Then Alice places a white stone at G. Then Bob places a black stone at I. Then Alice places a white stone at H. Then Bob places a black stone at K. Then Alice places a white stone at J. Where should Bob play next?
F
true
win
Bob
...*------W------W ../|...../|...../| ./.|..../.|..../.| *------B------B..| |..W---|--W...|..* |./....|./....|./ |/.....|/.....|/ *------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
58
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at C. Then Alice places a white stone at D. Then Bob places a black stone at G. Then Alice places a white stone at J. Then Bob places a black stone at I. Then Alice places a white stone at L. Where should Bob play next?
K
true
win
Bob
...W------B------W ../|...../|...../| ./.|..../.|..../.| W------*------B..| |..*---|--B...|..W |./....|./....|./ |/.....|/.....|/ *------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
59
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at A. Then Alice places a white stone at G. Then Bob places a black stone at C. Then Alice places a white stone at J. Then Bob places a black stone at E. Then Alice places a white stone at K. Where should Bob play next?
D
true
fork
Bob
...*------B------W ../|...../|...../| ./.|..../.|..../.| B------W------*..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
60
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at G. Then Bob places a black stone at E. Then Alice places a white stone at I. Then Bob places a black stone at F. Then Alice places a white stone at K. Where should Bob play next?
B
true
fork
Bob
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ B------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
61
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at B. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at G. Then Bob places a black stone at L. Then Alice places a white stone at J. Where should Bob play next?
K
true
fork
Bob
...W------*------W ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
62
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at E. Then Alice places a white stone at I. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
F
true
fork
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
63
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at E. Then Alice places a white stone at D. Then Bob places a black stone at F. Then Alice places a white stone at I. Then Bob places a black stone at L. Then Alice places a white stone at J. Where should Bob play next?
G
true
fork
Bob
...W------*------W ../|...../|...../| ./.|..../.|..../.| W------*------W..| |..*---|--*...|..B |./....|./....|./ |/.....|/.....|/ B------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
64
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at F. Then Alice places a white stone at E. Then Bob places a black stone at G. Then Alice places a white stone at I. Then Bob places a black stone at H. Then Alice places a white stone at J. Where should Bob play next?
B
true
fork
Bob
...W------*------W ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..B---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
65
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at F. Then Alice places a white stone at J. Then Bob places a black stone at I. Then Alice places a white stone at L. Where should Bob play next?
B
true
fork
Bob
...W------*------W ../|...../|...../| ./.|..../.|..../.| B------*------B..| |..W---|--*...|..W |./....|./....|./ |/.....|/.....|/ *------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
66
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at D. Then Alice places a white stone at H. Then Bob places a black stone at F. Then Alice places a white stone at I. Then Bob places a black stone at G. Then Alice places a white stone at L. Where should Bob play next?
C
true
fork
Bob
...B------*------* ../|...../|...../| ./.|..../.|..../.| W------*------W..| |..W---|--B...|..W |./....|./....|./ |/.....|/.....|/ *------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
67
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at C. Then Alice places a white stone at D. Then Bob places a black stone at F. Then Alice places a white stone at E. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
J
true
fork
Bob
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------W------*..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
68
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at B. Then Alice places a white stone at H. Then Bob places a black stone at F. Then Alice places a white stone at J. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
A
true
fork
Bob
...*------W------W ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ *------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
69
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at G. Then Alice places a white stone at C. Then Bob places a black stone at H. Then Alice places a white stone at E. Then Bob places a black stone at I. Where should Alice play next?
A, D, F
false
fork
Alice
...*------W------* ../|...../|...../| ./.|..../.|..../.| *------W------B..| |..B---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
70
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at F. Then Bob places a black stone at H. Then Alice places a white stone at G. Then Bob places a black stone at I. Then Alice places a white stone at L. Then Bob places a black stone at K. Where should Alice play next?
B, C, J
false
fork
Alice
...*------*------* ../|...../|...../| ./.|..../.|..../.| *------*------B..| |..B---|--W...|..W |./....|./....|./ |/.....|/.....|/ *------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
71
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at G. Then Bob places a black stone at E. Then Alice places a white stone at K. Then Bob places a black stone at F. Then Alice places a white stone at L. Then Bob places a black stone at J. Where should Alice play next?
B, C, I
false
fork
Alice
...*------*------B ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..*---|--W...|..W |./....|./....|./ |/.....|/.....|/ B------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
72
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at E. Then Bob places a black stone at I. Then Alice places a white stone at G. Then Bob places a black stone at L. Where should Alice play next?
A, F, H
false
fork
Alice
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------*------B..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------*------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
73
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at E. Then Alice places a white stone at B. Then Bob places a black stone at H. Then Alice places a white stone at F. Then Bob places a black stone at K. Where should Alice play next?
C, D, G
false
fork
Alice
...*------*------* ../|...../|...../| ./.|..../.|..../.| W------W------*..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
74
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at A. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at H. Then Bob places a black stone at K. Where should Alice play next?
B, D, E
false
fork
Alice
...*------W------* ../|...../|...../| ./.|..../.|..../.| B------*------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ *------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
75
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at A. Then Alice places a white stone at G. Then Bob places a black stone at D. Then Alice places a white stone at J. Then Bob places a black stone at F. Where should Alice play next?
C, I, L
false
fork
Alice
...B------*------W ../|...../|...../| ./.|..../.|..../.| B------W------*..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ *------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
76
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at A. Then Alice places a white stone at J. Then Bob places a black stone at F. Then Alice places a white stone at K. Then Bob places a black stone at H. Where should Alice play next?
C, I, L
false
fork
Alice
...*------*------W ../|...../|...../| ./.|..../.|..../.| B------W------*..| |..B---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
77
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at A. Then Alice places a white stone at F. Then Bob places a black stone at E. Then Alice places a white stone at G. Then Bob places a black stone at I. Where should Alice play next?
B, C, H
false
fork
Alice
...W------*------* ../|...../|...../| ./.|..../.|..../.| B------*------B..| |..*---|--W...|..* |./....|./....|./ |/.....|/.....|/ B------W------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
78
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at D. Then Alice places a white stone at C. Then Bob places a black stone at J. Then Alice places a white stone at H. Then Bob places a black stone at K. Where should Alice play next?
B, E, G
false
fork
Alice
...B------W------B ../|...../|...../| ./.|..../.|..../.| W------*------*..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------*------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
79
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at B. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at F. Then Alice places a white stone at K. Where should Bob play next?
C, D, G
false
fork
Bob
...*------*------W ../|...../|...../| ./.|..../.|..../.| B------W------*..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ B------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
80
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at C. Then Bob places a black stone at B. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at H. Then Bob places a black stone at I. Then Alice places a white stone at K. Where should Bob play next?
G, J, L
false
fork
Bob
...*------W------* ../|...../|...../| ./.|..../.|..../.| *------B------B..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ W------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
81
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at C. Then Alice places a white stone at H. Then Bob places a black stone at F. Then Alice places a white stone at I. Then Bob places a black stone at K. Then Alice places a white stone at J. Where should Bob play next?
B, G, L
false
fork
Bob
...*------B------W ../|...../|...../| ./.|..../.|..../.| W------*------W..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
82
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at E. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at K. Then Bob places a black stone at H. Then Alice places a white stone at L. Where should Bob play next?
A, B, C
false
fork
Bob
...W------*------* ../|...../|...../| ./.|..../.|..../.| *------*------*..| |..B---|--B...|..W |./....|./....|./ |/.....|/.....|/ B------W------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
83
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at A. Then Alice places a white stone at H. Then Bob places a black stone at C. Then Alice places a white stone at I. Then Bob places a black stone at F. Then Alice places a white stone at K. Where should Bob play next?
B, E, G
false
fork
Bob
...W------B------* ../|...../|...../| ./.|..../.|..../.| B------*------W..| |..W---|--*...|..* |./....|./....|./ |/.....|/.....|/ *------B------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
84
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at B. Then Alice places a white stone at F. Then Bob places a black stone at G. Then Alice places a white stone at H. Then Bob places a black stone at K. Then Alice places a white stone at J. Where should Bob play next?
C, I, L
false
fork
Bob
...*------*------W ../|...../|...../| ./.|..../.|..../.| W------B------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ *------W------B
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
85
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at E. Then Bob places a black stone at B. Then Alice places a white stone at H. Then Bob places a black stone at D. Then Alice places a white stone at I. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
A, C, F
false
fork
Bob
...B------*------* ../|...../|...../| ./.|..../.|..../.| *------B------W..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
86
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at C. Then Alice places a white stone at E. Then Bob places a black stone at F. Then Alice places a white stone at G. Then Bob places a black stone at L. Then Alice places a white stone at I. Where should Bob play next?
B, J, K
false
fork
Bob
...W------B------* ../|...../|...../| ./.|..../.|..../.| *------*------W..| |..*---|--W...|..B |./....|./....|./ |/.....|/.....|/ W------B------*
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
87
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at A. Then Bob places a black stone at B. Then Alice places a white stone at E. Then Bob places a black stone at G. Then Alice places a white stone at I. Then Bob places a black stone at J. Then Alice places a white stone at K. Where should Bob play next?
C, F, L
false
fork
Bob
...*------*------B ../|...../|...../| ./.|..../.|..../.| W------B------W..| |..*---|--B...|..* |./....|./....|./ |/.....|/.....|/ W------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
88
Alice and Bob are playing a game on two adjacent cubes. ABCD forms the top rectangle in the first cube and BIJC forms the top rectangle in the second cube. EFGH forms the bottom rectangle in the first cube and FKLG forms the bottom rectangle in the second cube. AE is an edge, BF is an edge, CG is an edge, DH is an edge, IK is an edge, and JL is an edge. Note that vertices B,C,G,F are shared by the two cubes. Alice and Bob plays a game where they take turns to put stickers on the vertices of the cubes that have not been occupied. Alice plays white stickers. Bob plays black stickers. The person who has four stickers on the same plane on either cube wins. Alice first places a white stone at D. Then Bob places a black stone at B. Then Alice places a white stone at H. Then Bob places a black stone at E. Then Alice places a white stone at J. Then Bob places a black stone at G. Then Alice places a white stone at K. Where should Bob play next?
A, C, F
false
fork
Bob
...W------*------W ../|...../|...../| ./.|..../.|..../.| *------B------*..| |..W---|--B...|..* |./....|./....|./ |/.....|/.....|/ B------*------W
...D------C------J ../|...../|...../| ./.|..../.|..../.| A------B------I..| |..H---|--G...|..L |./....|./....|./ |/.....|/.....|/ E------F------K
89

TTT-Bench: A Benchmark for Evaluating Reasoning Ability with Simple and Novel Tic-Tac-Toe-style Games

πŸ“ƒ Paper | πŸ€— Dataset | 🌐 Website

We introduce TTT-Bench, a new benchmark specifically created to evaluate the reasoning capability of LRMs through a suite of simple and novel two-player Tic-Tac-Toe-style games. Although trivial for humans, these games require basic strategic reasoning, including predicting an opponent's intentions and understanding spatial configurations. TTT-Bench is a simple and diverse benchmark consisting of four types of two-player games, namely oTTT (ordinary TTT), dTTT (double TTT), cTTT (TTT in cube), and sTTT (TTT with squares) (see Figure below for examples), and the questions in the benchmark test the reasoning capability of LRMs by posing the question of predicting the best move by a player. The objectives of these games are not only simple but are also introduced here for the first time (except oTTT), with no documented prior literature available online, making our benchmark uncontaminated and highly reliable.

Figure 1: Visualization of four game types in TTT-Bench (oTTT, dTTT, cTTT, and sTTT). For each game, the illustration consists of the board position annotations (above), the current game state with Alice's (white) and Bob's (black), and the correct next move (represented by a check mark).

Briefly, for each task in TTT-Bench, two players play a strategic game on a configuration dependent on the task to reach a winning state that satisfies a task-dependent winning constraint, and the player who reaches the winning state first wins the game. All the problems in this benchmark are formulated as a text-based board game question that starts with the description of the game, as well as the current game state with all the moves played by both players. The final TTT-Bench task questions are formulated to predict the next best move position by the current player, like: Where should Bob play next?.

Figure 2: Distribution of TTT-Bench question solution types. Above: Distribution of questions and their game verdicts (Win, Blocked, or Fork) based on how the game concludes after the corresponding solution move is played. Below: Distribution of questions and their solution types (single answer or multiple next-best moves).

The final TTT-Benchtest set consists of ~100 samples from each game. As shown in Figure above, not only is this dataset diverse across different game verdicts, but it also consists of questions with both single and multiple next-best move solutions. Overall, the final TTT-Bench benchmark consists of 412 questions from four simple two-player TTT games.

Dataset Details

The benchmark dataset consists of four splits corresponding to the four types of games, where in each split the dataset columns represents:

  • problem: Prompt question for the game.
  • solution: Ground truth next best move solution(s).
  • is_unique_solution: Boolean indicating whether there exists a single or multiple next best move solution(s).
  • verdict: Game verdict for the ground truth solution.
  • currect_board_visualization: Game board ASCII visualization with played moves.
  • board_visualization: Game board ASCII visualization with position labels.
  • problem_number: Problem ID

License

For fostering future research efforts in evaluating the reasoning capability of LRMs, we release this benchmark under the Apache 2.0 License.

Copyright(c) 2025 Advanced Micro Devices,Inc. All rights reserved.

Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at

http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

Citation

@misc{mishra2025tttbenchbenchmarkevaluatingreasoning,
      title={TTT-Bench: A Benchmark for Evaluating Reasoning Ability with Simple and Novel Tic-Tac-Toe-style Games}, 
      author={Prakamya Mishra and Jiang Liu and Jialian Wu and Xiaodong Yu and Zicheng Liu and Emad Barsoum},
      year={2025},
      eprint={2506.10209},
      archivePrefix={arXiv},
      primaryClass={cs.CL},
      url={https://arxiv.org/abs/2506.10209}, 
}
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