How many congruent regular tetrahedra can touch at a point

Definitions and mathematical statement

A regular tetrahedron is the closed convex hull of four points in $\mathbb R^3$ at equal pairwise distances. Fix one such tetrahedron $T$; congruent copies are its rigid-motion images, with no scaling. A conventional nonoverlapping common-point formulation seeks copies $T_1,\ldots,T_N$ and $p\in\mathbb R^3$ such that \[ p\in\bigcap_{i=1}^{N}\partial T_i,\qquad \operatorname{int}(T_i)\cap\operatorname{int}(T_j)=\varnothing \quad(i\ne j). \] Boundary contacts away from $p$ are allowed by this particular formulation.

Question. What is the largest number of congruent regular tetrahedra that can touch at one common point? The source reports an arrangement with at least $20$ tetrahedra and an upper bound $22$, and asks for the exact value. Its research summary repeats \[ 20\leq N_{\max}\leq22 \] and describes the upper bound as a simple volume bound, without providing a construction or proof.

The words touch at a point do not specify whether $p$ must be a vertex of every tetrahedron, whether contacts along additional edges or faces are allowed, or whether the pairwise intersections must equal $\{p\}$. Those stricter formulations are different from the displayed boundary-contact condition and are not silently imposed. The intended packing model must be matched to the reported bounds. There is no distinguished central tetrahedron: all copies are required to meet the same point, unlike a general central-body kissing-number problem. The supplied status remains open/triage rather than a verified exact value.

Short English statement

Find the maximum number of nonoverlapping congruent regular tetrahedra meeting at one point, between the reported bounds $20$ and $22$. The source needs an exact common-point and boundary-contact convention.

Sources

• Cooper, Combinatorial Problems I Like, Geometry, source-order bullet 10, October 2020 snapshot, https://people.math.sc.edu/cooper/combprob.html; HTML extraction, accessed 2026-07-29. Later resolution was not checked item by item according to the record.
• The supplied summary repeats the bounds $20$--$22$, mentions unspecified recent surveys, and contrasts a cube analogue with value $8$, but gives no separate construction, upper-bound, or survey citation.
• Local registry data/unsolvedmath/problems.json, permanent ID $3100010$, source code catalogue entry, How many congruent regular tetrahedra can touch at a point. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.