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If n = (k +1)q + r with 0 < r < k + 1, then the first person can win by removing r checkers each time. (Note r > 0 every time at the first person's turn since in the beginning it is so and the second person starts with a multiple of k + 1 checkers each time and can only remove 1 to k checkers.) However, if n is a multiple of k + 1, then no matter how many checkers the first person takes, the secon
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