Baker's Dozen — Relations among triangle areas

Definitions and mathematical statement

Fix a combinatorial type $T$ of a dissection of a square into $n\geq1$ labelled triangles. A realization consists of a square partitioned into triangles with pairwise disjoint interiors and with the prescribed combinatorial data; let $a_i>0$ be the Euclidean area of triangle $i$. Write $\mathcal A_T$ for the resulting set of ordered area vectors $(a_1,\ldots,a_n)$. The precise incidence data and allowed degenerations constituting $T$ are not specified in the supplied record.

The stated premise is that there is a nonzero polynomial relation, depending only on $T$, satisfied by every vector in $\mathcal A_T$. The problem asks what can be said about this relation and how its least degree can be determined from $T$. For example, once a coefficient field and realization convention have been chosen, polynomial relations can be expressed as \[ I_T=\{f\in{\mathbb{Q}}[x_1,\ldots,x_n]: f(a_1,\ldots,a_n)=0 \text{ for all }(a_1,\ldots,a_n)\in\mathcal A_T\}. \] A possible homogeneous version asks for \[ \min\{\deg f:0\ne f\in I_T,\ f\text{ homogeneous of positive degree}\}. \] This displays the distinction that must be resolved; it does not silently identify the source's "least degree" with this version.

In particular, if the square has fixed area $A$, every realization satisfies $\sum_i a_i=A$, an affine relation of degree one. The source's discussion of a distinguished irreducible homogeneous relation concerns a more specific area-polynomial formulation, whose normalization and treatment of this area-sum relation are not fully given. Allowing the square to scale, using normalized areas, and fixing a square lead to different ambient formulations. The intended relation ideal and degree notion must therefore be established before a unique algorithmic target is determined.

Short English statement

Determine the universal algebraic constraints on triangle areas for a prescribed square-dissection combinatorics, and compute the smallest degree of the intended relation from that combinatorics.

The registry labels the problem solved and its truncated research summary describes a distinguished irreducible homogeneous integer polynomial $p_T$, unique up to scalar, called the Monsky polynomial. It reports deformation invariance, a normalization with leading coefficients $\pm1$ in the parallelogram setting, and \[ p_T\equiv(x_1+\cdots+x_n)^d\pmod2,\qquad d=\deg p_T, \] as explaining Monsky's odd/even equidissection phenomenon. It also reports a principal relation ideal, a combinatorial degree lower-bound algorithm with exact worked examples, finiteness and explicit low-width relations denoted $\mathcal E_l$ for $l\leq4$, and direct enumeration for modest triangulation sizes. These are attributed claims of the supplied summary, not verified theorems asserted here: the numbered papers, width notation, normalization conventions, and complete final qualification are absent. The summary ends mid-sentence. The original 2015 source presents an open question, and the supplied status metadata explicitly requires review.

Sources

• UnsolvedMath permanent ID 600015, source code catalogue entry, "Baker's Dozen --- Relations among triangle areas." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Tabachnikov, A Baker's Dozen of Problems (2015), Problem 7, supplied publisher HTML locator https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/, accessed 2026-07-29 through an AMR-indexed list. Status evidence and rights review are marked NEEDS_REVIEW in the record.
• The research summary refers to "paper 1" through "paper 5" for its Monsky-polynomial and algorithmic claims, but supplies no authors, titles, or locators for those numbered references. No bibliography is invented for them.