Winning Strategy for a Card Game with Overlapping Cards

We claim that the player who starts, i.e., P1, has a winning strategy.

First, note that if there are only 1, 2, or 3 cards left on the table, the player who moves last will win by taking all the remaining cards. Therefore, we focus on the case where there are at least 4 cards left.

We will prove the claim by induction on the number of cards n. The base case of n = 4 is easy to check: P1 can take any two neighboring cards, and no matter what P2 does, P1 will be able to take the remaining cards and win.

Now suppose the claim holds for all values of n up to some positive integer k, and consider the case where there are k+1 cards on the table. If P1 takes one card, the remaining cards form a circle of k cards, and by the induction hypothesis, P2 will have a winning strategy. Therefore, P1 should not take one card.

If P1 takes two neighboring cards, the remaining cards form a circle of k-1 cards, and by the induction hypothesis, P2 will have a winning strategy. Therefore, P1 should not take two neighboring cards.

If P1 takes three neighboring cards, the remaining cards form a circle of k-2 cards, and by the induction hypothesis, P2 will have a winning strategy. Therefore, P1 should not take three neighboring cards.

At this point, P1 has only one option left: take any two non-neighboring cards. This will split the remaining cards into two circles, one with m cards and one with k+1-m cards, where m is at most (k+1)/2. P2 must then take one or two cards from one of these circles. Without loss of generality, assume P2 takes one card from the circle with m cards. Then the two circles have sizes m-1 and k+1-m.

Now P1 should take enough cards from the circle with size m-1 to make the two circles have equal sizes. This is always possible, since P1 can take at most two cards from this circle, and m-1 is at most (k+1)/2. After this move, the two circles have size (k+1)/2, and P2 will be forced to take one or two cards from one of the circles.

At this point, P1 can use the same strategy as in the base case of n = 4: take any two neighboring cards from the circle that P2 did not take from, and no matter what P2 does, P1 will be able to take the remaining cards and win.

Therefore, P1 has a winning strategy for all values of n, and should choose to play as P1.