## 2. Marel

Take a piece of cardboard and draw the following diagram on it:

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-097.jpg?height=368&width=651&top_left_y=267&top_left_x=725)

Also, take three silver and three gold coins (if you are on the beach, draw the diagram directly in the sand and play with three white and three black pebbles).

Now choose a suitable partner; each player starts the game with three coins (one player with gold, the other with silver). The player who starts places one of their coins on any of the nine circles. Then the second player places their coin on any of the remaining eight circles; after that, the first player places their second coin, and so on. When all the coins are placed, the players take turns moving one of their three coins along one of the sixteen segments, provided that the other end of the segment is free. The player who first aligns their three coins in a straight line wins.

I. After playing several rounds, you will undoubtedly notice that the player who starts the game can always win if they follow the appropriate strategy. What does it consist of?

II. Let's introduce an additional rule: the player who starts is not allowed to place their first coin in the center. In this case, the second player usually places their coin in the center. Then the first player has six different possible strategies:

1) place the first two coins at the midpoints of two opposite sides of the diagram;
2) place them at the midpoints of two adjacent sides of the diagram;
3) place one coin in a corner and the second in the midpoint of one of the opposite sides to that corner;
4) place one coin in a corner and the second in the midpoint of one of the sides converging at that corner;

${ }^{1}$ If you don't have gold coins, you can use copper ones,

5) place two coins in two opposite corners of the diagram;
6) place two coins in two adjacent corners.

Which of these strategies do you think is the best?

Historical information: Marel (or Merel) is a very ancient game, widely spread in the East under the name "game of three roads." The first definite mention of it appears in the works of Ovid ("Tristia" and "Ars Amatoria"). There it is said: "This game is played on a special small board with three coins that each player has; to win, one must align their three coins in a straight line."
2. Let's renumber our nine circles (nine positions of the scheme):

Let $C$ be the player who starts, and $F$ be the player who finishes placing their coins.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-226.jpg?height=363&width=646&top_left_y=544&top_left_x=705)

I. The player who starts the game places a coin in the center. First case. $F$ places their first coin in a corner. Then the game will develop as follows:

| $C$ | 5 | 8 | 3 | $8 \rightarrow 9$ | $5 \rightarrow 6$ | Victory! |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 1 | 2 | 7 | - | - |  |

Initial placement of 3 coins
Regular moves from 8 to 9, from 5 to 6, etc. ...

Second case. $F$ places their first coin in the middle of a side. The game develops as follows:

| $C$ | 5 | 1 | 7 | $1 \rightarrow 2$ | $2 \rightarrow 3$ | Victory! |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 8 | 9 | 4 | - | - |  |

In both cases, $C$ wins.

II. The starting player is prohibited from placing their first coin in the center.

Let's analyze the six game tactics listed in the problem statement.

1) The game develops as follows:

| $C$ | 2 | 8 | 1 | - |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 9 | 4 | $9 \rightarrow 6$ | Victory! |

2) The game develops as follows:

| $C$ | 2 | 6 | 3 | - |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 7 | 9 | $5 \rightarrow 8$ | Victory! |

3) The game develops as follows:

| $C$ | 1 | 6 | 3 | - |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 7 | 2 | $7 \rightarrow 8$ | Victory! |

4) The game develops as follows:

| $C$ | 1 | 2 | 7 | - |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 3 | 9 | $5 \rightarrow 6$ | Victory! |

5) The game develops as follows:

| $C$ | 1 | 9 | 2 | - |  |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 8 | 3 | $8 \rightarrow 7$ | Victory! |

6) The game develops as follows:

| $C$ | 1 | 3 | 8 | Draw |
| :---: | :---: | :---: | :---: | :---: |
| $F$ | 5 | 2 |  | Draw |

Thus, for the player $C$, who starts the game, the sixth tactic (placing the first two coins in two adjacent corners) is the best, as it is the only one that does not lead to a loss for $C$ (if the second player $F$ plays optimally), but to a draw.