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Omni-MATH / Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered…

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problem

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair (x,y)(x, y) of positive integers such that x≤20x \leq 20 and y≤23y \leq 23. (Philena knows that Nathan's pair must satisfy x≤20x \leq 20 and y≤23y \leq 23.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair (a,b)(a, b) of positive integers and tells it to Nathan; Nathan says YES if x≤ax \leq a and y≤by \leq b, and NO otherwise. Find, with proof, the smallest positive integer NN for which Philena has a strategy that guarantees she can be certain of Nathan's pair after at most NN rounds.
Plain-text mathematical notation (without MathML)
Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair (x,y) of positive integers such that x≤20 and y≤23. (Philena knows that Nathan's pair must satisfy x≤20 and y≤23.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair (a,b) of positive integers and tells it to Nathan; Nathan says YES if x≤a and y≤b, and NO otherwise. Find, with proof, the smallest positive integer N for which Philena has a strategy that guarantees she can be certain of Nathan's pair after at most N rounds.
Original LaTeX notation
Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan's pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan's pair after at most $N$ rounds.

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Source and history

task_id
9509aceb-3207-5b3b-a624-a7f74e10b60c
task_key
test--9509aceb-3207-5b3b-a624-a7f74e10b60c
task_revision_id
1
config / split
/ test

Official source

initial import