# The Domino Game

Given 28 stones of a domino game - on each stone is a pair of numbers, each number is between 0 and 6. Each possible pair of numbers appears on a separate stone.

[0|0] [0|1] [0|2] [0|3] [0|4] [0|5] [0|6]

[1|1] [1|2] [1|3] [1|4] [1|5] [1|6] [2|2]

[2|3] [2|4] [2|5] [2|6] [3|3] [3|4] [3|5]

[3|6] [4|4] [4|5] [4|6] [5|5] [5|6] [6|6]

Two players play against each other. The pool of stones is placed next to the players, Each player in turn takes a stone of his choice from the remaining stone pool. The first player takes the first stone and places it, And now each player in turn extends the row of dominoes to the right or left at will according to the legality of a domino game (That is, the player places a stone whose number matches to the number on a stone at the end of the line). The winning player is the player who, after his turn, the other palyer cannot extend the line, Because there is no suitable stone left for extending the row or no stones remains at all in the pool.

What strategy for winning the game could be developed?

• What are your thoughts on the matter? Where are you having trouble figuring it out? As a hint, note that each number occurs $8$ times on the stones, and except for the first domino, each placement of a domino removes two instances of some number from play. There is one more key observation needed to come up with a strategy, but I am going to leave it to you to figure out what that is. Dec 17, 2019 at 2:04

Perhaps not surprisingly, what appears to be crucial is the play of doubles.There is a simple winning strategy for the first player.

If possible play a domino whose open end has a number whose double has already been played.

Otherwise, play a double.

Proof

By symmetry, we can suppose the position after 3 plays is $$00|01|11.$$

Thereafter, when the second player plays $$0Y$$ or $$1Y$$, this is countered by the play of $$1Y$$ or $$0Y$$.

(Note that the dominoes with both numbers greater than 1 are never used and this strategy works for dominoes with numbers from $$0$$ to any $$n$$.)