# Who wins in $2$-player game on a $4 × 11$ chocolate bar? [closed]

A chocolate bar with rectangular grid has size 4 × 11. Two players make moves in turns. In each step a player breaks one of the current chocolate pieces along a grid line. The player who cannot make a step loses the game. Who wins - the player who starts the game or his opponent?

We consider a two-player game. The players are given an x × y rectangle (where x and y are positive integers). The players take turns moving. A move consists of dividing a rectangle into two rectangles with a single vertical or horizontal cut. The resulting rectangles must have positive integer dimensions.

• When you write "the grid line" in the third sentence do you mean "a grid line"? – Qiaochu Yuan Jan 17 at 0:36
• You can solve this problem algorithmically for any $n$ and $m$ using the theory of Grundy numbers en.m.wikipedia.org/wiki/Nimber. – vujazzman Jan 17 at 0:46
• @QiaochuYuan exactly. That's what I meant. – Muhammed Abdulkabir Jan 17 at 0:47

When you break along a grid line, you increase the number of separate pieces of chocolate by $$1$$. You start off with one piece of chocolate, and at the end of the game there are $$xy$$ separate pieces.
So player $$1$$ wins when $$xy$$ is even, and player $$2$$ wins when $$xy$$ is odd.