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Statement

Let $T=(A,B,C)$ have incommensurable angles, with $C=A/2+B$ and $s=2\sin(A/4)=M/K\in\mathbb Q$, where $M,K$ are positive integers. Then $T$ has a tiling by the triangle $R=(\alpha,\beta,\gamma)$ with $\alpha=A/2$ and $\beta=B$. For every tiling by this shape, its count $N$ is square if and only if $2K^2-M^2$ is square.

The criterion is independent of the chosen representation of $s$. It does not assert that $2K^2-M^2$ itself is an attainable tile count for every representation $M/K$.

Source and scope. Beeson–Laczkovich–Zhang, arXiv:2604.03609v3, Proposition 29, pp. 15–16. Complete rewritten deduction from the external existence and counting results below. The extra minimum-count remarks on p. 16 are not used in this proof.

Proof

The angle sum gives $3\alpha+2\beta=\pi$, and $T=(2\alpha,\beta,\alpha+\beta)$. Since $\sin(\alpha/2)=s/2$ is rational, Laczkovich, Tilings of triangles (1995), Theorem 2.4, gives a tiling of $T$ by $R$.

For any such $N$-tiling, the required external counting theorem is Beeson, Triangle tiling: the case $3\alpha+2\beta=\pi$, arXiv:1206.2229v3, Theorem 4, p. 28. It supplies positive integers $m,k$ with

$$s=\frac mk,\qquad N=2k^2-m^2,\qquad m^2<N.$$

Since $m/k=M/K$, there is a positive rational $\lambda$ with $(m,k)=\lambda(M,K)$. Thus

$$N=\lambda^2(2K^2-M^2).$$

Multiplication by a nonzero rational square preserves being a rational square. Both $N$ and $2K^2-M^2$ are positive integers, and an integer that is a rational square is an integer square. This proves the criterion.

Source qualifications

The necessity proof of Theorem 1 on p. 9 cites “Theorem 4 of [3]” for this equation. Reference [3] there is the isosceles-triangle paper. The proof of Proposition 29 correctly cites reference [1], and the equation is indeed Theorem 4 of arXiv:1206.2229v3, p. 28. We use that identified input.

The additional p. 16 remark claims the necessity of $K\mid M^2$ from that same theorem and hence an exact smallest count. The cited theorem's statement and proof do not establish that divisibility; Theorem 5 of the external paper assumes it for a construction. This extra necessity and the claimed exact minimum require separate justification and are not asserted here. Neither is needed for the square criterion.

Bears on. Problem 633.