Odd rep-tiling by a 14-omino

Definitions and mathematical statement

Fix the $14$-omino \[ T=([2,6]\times[0,3])\ \cup\ ([0,2]\times[2,3])\subset{\mathbb{R}}^2. \] It is a $3\times6$ rectangle with a $2\times2$ corner removed, with its polygonal boundary retained. Thus $\operatorname{area}(T)=14$. Removing any other corner gives a congruent tile.

The question is whether there exist a positive odd integer $N$, real numbers $a,b>0$, and congruent copies $T_1,\ldots,T_N$ of $T$ such that \[ [0,a]\times[0,b]=\bigcup_{i=1}^N T_i, \qquad \operatorname{int}(T_i)\cap\operatorname{int}(T_j)=\varnothing \quad(i\ne j). \] For the literal Euclidean-congruence reading, each $T_i=g_i(T)$ for a plane isometry $g_i$; this allows rotations and reflections. The copies have the same scale as $T$. Necessarily the rectangle area is $ab=14N$; the rectangle's aspect ratio is not prescribed.

The source does not explicitly say whether the intended polyomino tilings must respect a common square lattice or whether reflections are disallowed. Those restrictions would define potentially different variants and are not silently added to the literal congruence formulation. Despite the source title's word "rep-tiling," the supplied question asks to tile a rectangle, not necessarily a larger similar copy of $T$.

Short English statement

Can this corner-notched $14$-square tile cover some rectangle without interior overlaps using an odd number of equal-sized congruent copies? The supplied research summary says no resolution for this specific shape was found and that general odd-reptile theory gives context without answering it. This is an OPEN-TRIAGE assessment. Provenance says the Geometry Junkyard index presents it as open but later item-level resolution was not checked.

Sources

• Local UnsolvedMath registry, permanent ID 3900010, source code catalogue entry, "Odd rep-tiling by a 14-omino." Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• David Eppstein, The Geometry Junkyard: Open Problems, item "Odd rep-tiling by a 14-omino," https://ics.uci.edu/~eppstein/junkyard/open.html; HTML extraction, accessed 2026-07-29. Attribution in the registry is to David Eppstein and cited proposers, without a separate original proposer or paper.
• The supplied research summary contains no additional primary citation for the specific shape or its reported unresolved status.
• Source provenance marks status evidence NEEDS_REVIEW and rights NEEDS_REVIEW: public access was verified, but redistribution permission was not established. This source-specific qualification is separate from the dataset licence.