benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Alice and Bob play a game in which they take turns choosing integers from 1 to n. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice ch…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Alice and Bob play a game in which they take turns choosing integers from 11 to nn. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the nn integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the nnth turn, which is forced and ends the game. Bob wins if the parity of $\{k\colon \mbox{the number $kwaschosenonthe was chosen on the k$th turn}\}$ matches his goal. For which values of nn does Bob have a winning strategy?
Plain-text mathematical notation (without MathML)
Alice and Bob play a game in which they take turns choosing integers from 1 to n. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the n integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the nth turn, which is forced and ends the game. Bob wins if the parity of $\{k\colon \mbox{the number $kwaschosenonthek$th turn}\}$ matches his goal. For which values of n does Bob have a winning strategy?
Original LaTeX notation
Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of "odd" or "even". On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\{k\colon \mbox{the number $k$ was chosen on the $k$th turn}\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
d84a2295-a8a4-5f5a-bb0a-a5ffbb727de0
task_key
test--d84a2295-a8a4-5f5a-bb0a-a5ffbb727de0
task_revision_id
1
config / split
/ test

Official source

initial import