20. A 'long knight' moves on a square grid. A single move, as shown, consists of moving three squares in one direction (horizontally or vertically) and then one square at right angles to the first direction. What is the smallest number of moves a long knight requires to go from one corner of an $8 \times 8$ square board to the diagonally opposite corner?

A 4
B 5
C 6
D 7
E 8
Solution: B
Method 1: Our first method is systematic but long winded. In the left hand diagram we have put 0 in the square where the long knight starts and $1 \mathrm{~s}$ in the squares which the long knight can reach in 1 move. In the middle diagram we have put a 2 in all the squares, other than that already marked 0 , that the long knight can reach in one move from the squares marked 1. The nine squares marked with a 2 in this diagram are all the squares that the long knight can reach in two moves.
Next we put a 3 in all the squares, other than those already marked, which the long knight can reach in one move from the squares marked with a 2 , and so on. The completed diagram is shown on the right. The number in a square indicates the smallest number of moves it takes the long knight to reach that square. We see from this that it takes the long knight five moves to reach the corner which is diagonally opposite the square where it started.
Method 2: We note that to get from the bottom left-hand corner to the top right-hand corner the long knight must end up 7 squares up and 7 squares right from its original position, a total of 14 squares. In any one move, it goes at most 3 squares up and 1 squares to the right, or at most 1 square up and 3 squares to the right, in each case a total of 4 squares.
So after 3 or fewer moves it cannot end up in the top right-hand square. In 4 moves the number of squares it will have moved upwards must be even, as it will be the sum of four odd numbers, and so cannot be the required 7 squares to reach the top right-hand corner. Therefore at least 5 moves are needed. The diagram shows one way in which the task may be achieved in 5 moves.