benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / Let Z^(n) be the integer lattice in R^(n). Two points in Z^(n) are called …1 in one coordinate and are equal in all other coordinates. For which integers n≥1 does there exist a set…

Problem

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

problem

Let Zn\mathbb{Z}^n be the integer lattice in Rn\mathbb{R}^n. Two points in Zn\mathbb{Z}^n are called \emph{neighbors} if they differ by exactly 11 in one coordinate and are equal in all other coordinates. For which integers n≥1n \geq 1 does there exist a set of points S⊂ZnS \subset \mathbb{Z}^n satisfying the following two conditions? \begin{enumerate} \item[(1)] If pp is in SS, then none of the neighbors of pp is in SS. \item[(2)] If p∈Znp \in \mathbb{Z}^n is not in SS, then exactly one of the neighbors of pp is in SS. \end{enumerate}
Plain-text mathematical notation (without MathML)
Let Z^(n) be the integer lattice in R^(n). Two points in Z^(n) are called \emph{neighbors} if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers n≥1 does there exist a set of points S⊂Z^(n) satisfying the following two conditions? \begin{enumerate} \item[(1)] If p is in S, then none of the neighbors of p is in S. \item[(2)] If p∈Z^(n) is not in S, then exactly one of the neighbors of p is in S. \end{enumerate}
Original LaTeX notation
Let $\mathbb{Z}^n$ be the integer lattice in $\mathbb{R}^n$. Two points in $\mathbb{Z}^n$ are called \emph{neighbors} if they differ by exactly $1$ in one coordinate and are equal in all other coordinates. For which integers $n \geq 1$ does there exist a set of points $S \subset \mathbb{Z}^n$ satisfying the following two conditions? \begin{enumerate} \item[(1)] If $p$ is in $S$, then none of the neighbors of $p$ is in $S$. \item[(2)] If $p \in \mathbb{Z}^n$ is not in $S$, then exactly one of the neighbors of $p$ is in $S$. \end{enumerate}

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
ee33508b-a6ec-5b91-aa13-f417500d9537
task_key
test--ee33508b-a6ec-5b91-aa13-f417500d9537
task_revision_id
1
config / split
/ test

Official source

initial import