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Claim. Alexander Soifer, Is there anything beyond the solution?, chapter 6 of How Does One Cut a Triangle?, 2nd ed., Springer, New York, 2009, pp. 47–50, DOI 10.1007/978-0-387-74652-4_6, reports Erdős's question and shows that the numbers $2n^2$, $3n^2$, $6n^2$ and $n^2+m^2$ occur as the number of congruent triangles into which some triangle can be cut, beside the squares $n^2$, which any triangle realizes. This is the source the site's remarks on Problem 634 credit for these families, and the 2026 manuscripts on the prime values cite the same pages for the classical constructions; the page is dated to the year of the cited edition.

Covers. The values $2n^2$, $3n^2$, $6n^2$ and $n^2+m^2$ occur for all positive integers $n$ and $m$. Nothing is claimed about other values of $n$.

Depends on. No page of this wiki.

Standing. Claimed. The chapter is part of a monograph, not a refereed journal publication, and the site labels the problem OPEN, so its credit is commentary, not acceptance. The constructions are used as known by George, Harries, Beeson and Bonfioli in their manuscripts recorded on this problem's other claim pages.