Last Biscuit: Getting Started

If one stack contains one biscuit and the other stack contains two biscuits, the player to go next cannot possibly win. (We'll write this configuration as $(1,2)$.)

• If the first player takes away the biscuit from the stack containing only one biscuit, then only the 2-biscuit stack is left, which the second player can take away to win.
• If the first player takes away a biscuit from the stack containing two biscuits, then two stacks with one biscuit each are left, which the second player can take away to win.
• If the first player takes away both biscuits from the stack containing two biscuits, then then only the 1-biscuit stack is left, which the second player can take away to win.

Therefore, $(1,2)$ is a losing configuration for the player to go next.

One winning configuration is $(2,3)$. If you take one biscuit from each stack, your opponent is faced with the losing configuration $(1,2)$. Another is $(1,3)$. Taking one biscuit from the 3-stack also leaves your opponent with the losing position $(1,2)$.

There are infinitely many winning configurations from which you can reach $(1,2)$ in one move.

• Any configuration of the form $(1,n)$ where $n>2$. You simply take $n-2$ biscuits from the $n$-stack to leave your opponent with the losing configuration. (Note that a configuration $(n,m)$ offers the same play options as $(m,n)$ so we will treat these as the same.)
• Any configuration of the form $(2,n)$ where $n>1$. You simply take $n-1$ biscuits from the $n$-stack to leave your opponent with the losing configuration.
• Any configuration of the form $(n,n+1)$, where $n > 1$. You simply take $n-1$ biscuits from both stacks to leave your opponent with the losing configuration.

As we have just seen, configurations $(1,n)$, $(2,n)$ and $(n,n+1)$ for $n>1$ are ruled out because they are winning positions. The next configuration to consider is $(3,5)$. You can quite easily convince yourself that whatever the player to go first does will leave their opponent with a winning position. Therefore $(3,5)$ is a losing position for the player who goes first.