Baltic Way Shortlist, 2023

We make a table inductively that shows whether the player, whose turn it is if there are $n$ marbles left, wins (marked with a W) or loses (marked with an L).

The proof of the correctness of the table is inductive. If there are at most 35 marbles left, a player cannot remove a number of marbles of the type $6k$ with $k \in \mathbb{Z}$, because the numbers 6, 12, 18, 24 and 30 are neither prime numbers nor squares. We can conclude inductively that, as long as $n \le 35$, the number $n$ should be marked with an L if it is divisible by 6 (since a legal move will yield a number that is not divisible by 6), and it should be marked with a W if it is not divisible by 6 (since removing 1, 2, 3, 4 or 5 marbles yields to a number that is divisible by 6).

This argument is not valid for $n = 36$, as one can remove all 36 marbles in one step and thus win.

The number $n = 37$ is also winning, since one can remove 7 marbles to bring the opponent into a losing position with 30 marbles.

With $n = 38$ marbles left it is not possible to win, since one would need to remove $38 - 30 = 8$, $38 - 24 = 14$, $38 - 18 = 20$, $38 - 12 = 26$, $38 - 6 = 32$ or all 38 marbles to bring the opponent into a losing position, but the numbers 8, 14, 20, 26, 32 and 38 are neither prime numbers nor squares.

The next five numbers, 39, ..., 43 are winning positions, because a removal of 1, ..., 5 marbles leaves the opponent in the losing position with 38 marbles.

However, $n = 44$ is again a losing position, because one would need to remove $44 - 38 = 6$, $44 - 30 = 14$, $44 - 24 = 20$, $44 - 18 = 26$, $44 - 12 = 32$, $44 - 6 = 38$ or all 44 marbles to force the opponent into a losing position, but these moves are not allowed.

The next five numbers, 45, ..., 49 are winning positions, because a removal of 1, ..., 5 marbles leaves the opponent in the losing position with 44 marbles.

Now the number 50 is left and it is again a losing position, because Alfred would need to remove $50 - 44 = 6$, $50 - 38 = 12$, $50 - 30 = 20$, $50 - 24 = 26$, $50 - 18 = 32$, $50 - 12 = 38$, $50 - 6 = 44$ or all 50 marbles to bring Bodil into a losing position, but these moves are not allowed. Hence Bodil will win the game if she plays in an optimal way.