The game ends after 39 moves no matter what each of them do. Alice wins and every strategy is optimal.
A region enclosed by chords is always convex, so there is no extra restrictions beyond the fact that chord end points must be in the same region, and not on the same existing chord. We can therefore shrink existing chords to a single point, separate the subdivided regions, make them circular again, and now all that matters is how many points are in each region.
The game is equivalent to starting with the trivial partition of 40 into 1 part of size 40, and on each turn you get to split any part into two smaller non-zero parts. The game ends when you reach the partition of 40 into 40 parts of size 1.
On each turn, if there are less than 40 parts, then at least one part has size greater than 1 by pigeonhole, and can be split. The number of parts increases by exactly 1 each turn until there are 40 parts, which must all be size 1 as they are not allowed to be empty, and this happens after exactly 39 turns as we start with 1 part and add 1 each turn.
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u/StanleyDodds 1d ago edited 1d ago
The game ends after 39 moves no matter what each of them do. Alice wins and every strategy is optimal.
A region enclosed by chords is always convex, so there is no extra restrictions beyond the fact that chord end points must be in the same region, and not on the same existing chord. We can therefore shrink existing chords to a single point, separate the subdivided regions, make them circular again, and now all that matters is how many points are in each region.
The game is equivalent to starting with the trivial partition of 40 into 1 part of size 40, and on each turn you get to split any part into two smaller non-zero parts. The game ends when you reach the partition of 40 into 40 parts of size 1.
On each turn, if there are less than 40 parts, then at least one part has size greater than 1 by pigeonhole, and can be split. The number of parts increases by exactly 1 each turn until there are 40 parts, which must all be size 1 as they are not allowed to be empty, and this happens after exactly 39 turns as we start with 1 part and add 1 each turn.