Question 51
Player A and B play a painful game on the real line. Player A has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player A picks some positive integer $m$ and provides $\frac{1}{2^m}$ units of ink from the pot. Player B then picks an integer $k$ and blackens the interval from $\frac{k}{2^m}$ to $\frac{k+1}{2^m}$ (some parts of this interval may have been blackened before.) The goal of player A is to reach a situation where the pot is empty and the interval $[0, 1]$ is not completely blackened.
Decide whether there exists a strategy for player A to win in a finite number of moves.
Bilingual source note: the Chinese statement requires a positive integer k, whereas the English statement above allows any integer k. Both readings are analyzed below.
Decide whether there exists a strategy for player A to win in a finite number of moves.
Bilingual source note: the Chinese statement requires a positive integer k, whereas the English statement above allows any integer k. Both readings are analyzed below.