Strategy

The core idea is to find a strategy that guarantees success regardless of your opponent's choices or the hidden value of $N$. In games, this often means identifying "safe positions"—amounts that leave your opponent in a losing spot. For the stone games, if players can take $1$ through $k$ stones, multiples of $k+1$ are losing positions because any move your opponent makes can be countered to return the total to the next multiple of $k+1$. In the egg problem, you balance the risk of breaking the first egg with the need to test remaining floors linearly with the second egg to minimize the maximum number of drops.