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Race to call 200 with steps of one to eight

Question

Two players alternately call out numbers. The first player opens with any number from 11 to 88. After that, each call must exceed the previous call by at least 11 and at most 88. Whoever calls 200200 wins. Is there an opening number that guarantees the first player a win with correct play? If so, give it; if not, answer −1-1.

Short answer

22. Since every turn adds between 11 and 88, the first player can always restore the running total to a number congruent to 200(mod9)200 \pmod 9, i.e. of the form 9k+29k+2. Opening with 22 and then answering each move with the next 9k+29k+2 (11,20,…,19111, 20,\dots,191) leaves the opponent unable to stop 200200.

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