Let $n>2$ be an integer. Anna, Edda, and Magni play a game on a hexagonal board tiled with regular hexagons, with $n$ tiles on each side. The figure shows a board with 5 tiles on each side. The central tile is marked.

## Baltic Way

Reykjavík, November 11th - 15th

Solutions

The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda, and Magni take turns moving the stone: Anna begins, then Edda and then Magni, and so on.

The rules for each player's turn are:

- Anna has to move the stone to an adjacent tile, in any direction.

![](https://cdn.mathpix.com/cropped/2024_04_17_e27e300a283338a2e031g-08.jpg?height=386&width=425&top_left_y=495&top_left_x=1438)

- Edda has to move the stone straight by two tiles in any of the 6 possible directions.
- Magni has a choice of passing his turn, or moving the stone straight by three tiles in any of the 6 possible directions.

Find all $n$ for which Edda and Magni have a winning strategy.
We colour the board in three colours in such a way that no neighbouring tiles are of the same colour. We can give each hexagon a coordinate using $\overrightarrow{e_{1}}=(1,0)$ and $\overrightarrow{e_{2}}=\left(\cos \left(120^{\circ}, \sin \left(120^{\circ}\right)\right)=\right.$ $\left(\frac{-1}{2}, \frac{\sqrt{3}}{2}\right)$ as basis. Let the central tile be the origin. Then each hexagon has center at $a \cdot \overrightarrow{e_{1}}+b \cdot \overrightarrow{e_{2}},(a, b) \in$ $\mathbb{Z}^{2}$. The tuple $(a, b)$ is the coordinate for a given hexagon its neighbours are $(a+1, b),(a+1, b+1)$, $(a, b+1),(a-1, b),(a-1, b-1)$ and $(a, b-1)$. We colour the hexagon with coordinates $(a, b)$ with colour number $(a+b)(\bmod 3)$. It is clear that neighbouring hexagons do not share a colour. (In fact this is the only three colouring of a hexagonal tiling). See figure 3.

![](https://cdn.mathpix.com/cropped/2024_04_17_e27e300a283338a2e031g-08.jpg?height=437&width=502&top_left_y=1712&top_left_x=777)

Figure 3: Three colouring of the hexagonal tiling for 5 hexagons on each side

We see that if $n \equiv 1(\bmod 3)$, the stone begins in a tile in the same colour as the central tile, let that colour be grey. By regarding a few cases, we see that whatever Anna does, Edda and Magni can end

![](https://cdn.mathpix.com/cropped/2024_04_17_e27e300a283338a2e031g-09.jpg?height=180&width=363&top_left_y=133&top_left_x=181)

BALTIC WAY

REYKJAVÍK$\cdot$ 2021

## Baltic Way

Reykjavík, November 11th - 15th
Solutions

their turns by getting the stone to a prescribed grey tile of the closest grey tiles. Therefore they can get the stone to the central tile.

If $n \not \equiv 1(\bmod 3)$, the stone does not begin on the same grey colour as the central tile. Say the stone begins on a white tile, and say the third colour is black. Anna can always move the stone to a grey tile that is not on the same horizontal/diagonal line as the central tile. Then Anna moves the stone to a white or black tile. After Magni moves the stone is still again on a white/black tile. Anna can continue this indefinitely, with the stone never reaching the central tile.