Figurate and Polygonal Numbers

Figurate numbers count how many dots it takes to arrange a shape of a given size, growing that shape one ring or row at a time. The most familiar is the triangular numbers — 1, 3, 6, 10, 15 — formed by stacking rows of 1, 2, 3, 4, 5 dots into a triangle, giving the formula $T_n = n(n+1)/2$. Square numbers — 1, 4, 9, 16 — are just $n^2$, formed by an $n \times n$ grid of dots. More generally, the polygonal number for a $k$-sided regular polygon of "size" $n$ is given by $P(k,n) = \frac{(k-2)n^2 - (k-4)n}{2}$, which reduces to the triangular formula at $k=3$ and the square formula at $k=4$.

These show up in interviews less as "recite the formula" and more as counting puzzles in disguise: how many handshakes happen among $n$ people (triangular numbers, since it is choosing 2 out of $n$), or how many unit squares are in a triangular grid of side $n$. Recognizing that a count is secretly a figurate number turns a fiddly enumeration into a one-line closed form. A useful identity worth memorizing is that every square number is the sum of two consecutive triangular numbers: $T_{n-1} + T_n = n^2$, which follows directly from stacking two staircases of dots into a square.