Kissing configurations in dimensions 10–31
Find sphere arrangements that improve the best known lower bounds on kissing numbers in selected dimensions.
Cite
@misc{cairn-kissing-number-lower-bounds,
title = {Kissing configurations in dimensions 10–31},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/kissing-number-lower-bounds}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
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The problem
The kissing number τ_d is the maximum number of non-overlapping unit spheres in R^d that can touch a central unit sphere. It is known exactly only in a few dimensions (1–4, 8, 24). Elsewhere we know lower bounds from explicit configurations and upper bounds from semidefinite programming.
Submission format: a list of vectors with exact (integer or rational, possibly scaled) coordinates. The checker verifies all vectors have equal norm and all pairwise inner products are at most half the squared norm (angles ≥ 60°). Score = number of vectors for the stated dimension.
Score: number of vectors in a valid kissing configuration of the stated dimension (maximize)· checker kissing