Six, and the Two Triangle Dissection Problems

There are two natural but distinct questions about dissecting a triangle into congruent triangular pieces. In the first, the pieces must also be similar to the original triangle; in the second, no such similarity is required. The two problems coincide for the integers $1$ through $5$, and they first diverge at $6$. The first problem is the classical rep-$n$ problem. A triangle is rep-$n$ precisely when $n$ is a perfect square, a sum of two positive squares, or three times a square, so $6$ is the smallest integer admitting no rep-$n$ dissection at all. An equilateral triangle can nevertheless be dissected into six mutually congruent triangles: its three altitudes divide it into six $30^\\circ$-$60^\\circ$-$90^\\circ$ triangles, and none of these is similar to the original. The distinction rests on a single area identity. If $n$ similar copies of a triangle assemble into that triangle, their common similarity ratio is necessarily $1/\\sqrt{n}$. This identity supplies the arithmetic underlying the rep-$n$ classification and makes $6$ the first integer at which congruence alone becomes strictly weaker than congruence together with similarity to the parent.