# Given an $8×8$ square and a set, which contains the pentominoes and four $1×1$ squares

I have a game.

Given an $8 × 8$ square and a set, which contains the pentominoes and four $1 × 1$ squares. Players alternately pick one item from the set. Then players (starting with the player who had chosen the first item) take turns in placing pentominoes on the board so that they do not overlap with existing tiles and no tile is used more than once. The objective is to be the last player to place a tile on the board.

So I need a great strategy. I am sure there doesn’t exist a winning strategy (and if so, then it is complicated), so I only need a strategy which helps to win.

• The picture you've chosen to use is a bit misleading, as it shows the 18 one-sided pentominoes, where as you are using the 12 two-sided ones, i.e. you are allowed to turn them over before placing them on the board. Also just to confirm - you are including the 4 monominoes in the set of playable game pieces? Commented Jul 31, 2018 at 17:01

## 1 Answer

Pretty sure a winning strategy has to exist, either for the starting player or the opponent. There are a finite number of possible games, and there's no draw possible.

That said, I don't have a solution. Just some thoughts.

I'd probably start by looking at pieces whose placement is the most limited. E.g. if there is only one spot left on the board with five successive empty squares and the opponent has a long piece left, it's useful to block that spot to prevent them from playing that piece. Similarly, if we have a piece left which can only go in one spot on the board, we want to play that piece first in order to prevent the opponent from blocking the spot. By default I'd leave the 1x1 pieces last since they can be played on any board position, but there may be situations where it's useful to use them earlier (e.g. if we can block multiple opponent's pieces with a single 1x1).