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Partial answer (without a proof, only some assumptions): Explanation:


@Deusovi was a few hours earlier, but perhaps this answer gives a simpler argument for the lower bound and a more practical description of an optimal strategy. Let N (=52) be the number of cards and k (=6) the smallest integer such that $N \le 2^k$. First, let us show that at least k operations are required: Optimal algorithm:


The optimal strategy takes Here's why this is necessary: And here's a way to do it: Fun fact:


There are already a couple of answers, but I think this question is a bit better than just a straightforward calculation problem, so here's one more. You can reword the question into 5 separate questions like this: Here is a card of which you know nothing, except that it was the smallest in a random sample of 5 cards. Do you wish to exchange it for a card ...


I wasn't fast enough, but here's some code confirming Jerry's answer: from itertools import combinations wins = [0]*6 total = [0]*6 for A in combinations(range(10),5): rem = set(range(10))-set(A) for N in range(6): for draw in combinations(rem,N): total[N] += 1 if sum(A[N:]+draw) > 22: wins[N] += 1 for i in range(6):...


Here's my solution based on expectations. First, The calculation steps are shown below: Next, let's think about A's strategy. Then, In conclusion:

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