PUMaC 2021 · 个人决赛（A 组） · 第 3 题

PUMaC 2021 — Individual Finals (Division A) — Problem 3

Alice and Bob are playing a game, starting with a binary string b of length 2022. In each step, the rightmost digit of the string is deleted. If the deleted digit was 1, Alice gets to choose which digit she wants to append on the left. Otherwise, Bob gets to choose the digit to append on the left of the string. Alice would like to turn the string b into the all-zero string 00 . . . 0 , in the least number of | {z } 2022 steps possible, while Bob would like to maximize the number of steps necessary, or prevent Alice from doing this at all. a ) Is there a string b for which Bob can prevent Alice in her goal, if both players play optimally? b ) If the answer to part a is yes, find all such strings b . If the answer is no, find the maximal game time and find the set of strings b for which the game time is maximal. 1

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