On the Number of Distinct Tilings of Finite Subsets of Z^d With Tiles of Fixed Size

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Problem: Reconstructed conjecture: for $\alpha,n,d\in\mathbb Z^+$ and every finite $C\subset\mathbb Z^d$ with $|C|=n$,

where $[n]=\{1,\dots,n\}$, and $\mathcal T(\alpha,C)$ is the set of canonical translational tilings $(A,B)$ with $|A|=\alpha$, $A+B=C$, and $|C|=|A||B|$, counted up to translating $A$ and $B$ oppositely. This is the formal version stated in Stern’s paper as Conjecture $[n]\mathrm{isBest}$. The abstract wording is slightly ambiguous, but this is the precise surrounding formulation.

Result: The conjecture is false already for $d=1$, $\alpha=2$, $n=26$.

Let

Then $|C|=26$. We exhibit three distinct tilings of $C$ by two-point tiles:

Directly,

for $r=3,5,15$, and each $B_r$ contains $0$ and is nonnegative, so these are canonical tilings. Hence

It remains to check the benchmark interval. Translating $[26]$ to $I=\{0,\dots,25\}$ does not change the count. Suppose $(A,B)\in\mathcal T(2,I)$. Since $0\in B$ and $B\subseteq\mathbb Z_{\ge0}$, we must have $A=\{0,r\}$ for some $1\le r\le25$. Then

Along each residue class modulo $r$, the points of $I$ form a finite chain. The first point of each chain must lie in $B$, membership then alternates, and the last point cannot lie in $B$, since its translate by $r$ would leave $I$. Thus every chain length must be even.

Writing $26=qr+t$, $0\le t<r$, the residue chains have lengths $q$ and, if $t>0$, also $q+1$. Hence $t=0$ and $q=26/r$ must be even. Thus $r=1$ or $r=13$. Each such $r$ determines $B$ uniquely, so

Therefore

contradicting the conjecture.

Verification audit: the counterexample uses the paper’s own definition of $\mathcal T$, keeps $\alpha=2$, $n=26$, $d=1$, and compares a finite $C\subset\mathbb Z$ of size $26$ against $[26]$. No extra hypotheses are introduced.

Citation: Jesse Stern, “On the Number of Distinct Tilings of Finite Subsets of $\mathbb Z^d$ With Tiles of Fixed Size,” arXiv:2303.06717. The counterexample above is self-contained.