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Opening move in a race to fifty

Question

Two players take turns adding a whole number between 11 and 1010 (inclusive) to a running total that starts at 00. Whoever brings the total to exactly 5050 wins. You move first. Which number should you open with to guarantee a win?

Short answer

66. Whatever your opponent adds, you can reply so the two moves sum to 1111. So the winning totals are those congruent to 50≡6(mod11)50\equiv6\pmod{11}: 6,17,28,39,506,17,28,39,50. Open with 66 and keep landing on that sequence.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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