Baltic Way 2023 Shortlist

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).

$n$	1 W -1	2 W -2	3 W -3	4 W -4	5 W -5	6 L	7 W -1	8 W -2	9 W -3	10 W -4
$n$	11 W -5	12 L	13 W -1	14 W -2	15 W -3	16 W -4	17 W -5	18 L	19 W -1	20 W -2
$n$	21 W -3	22 W -4	23 W -5	24 L	25 W -1	26 W -2	27 W -3	28 W -4	29 W -5	30 L
$n$	31 W -1	32 W -2	33 W -3	34 W -4	35 W -5	36 W -36	37 W -7	38 L	39 W -1	40 W -2
$n$	41 W -3	42 W -4	43 W -5	44 L	45 W -1	46 W -2	47 W -3	48 W -4	49 W -5	50 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.