Adam Scherlis

Math facts: 2025

As you have no doubt heard by now,

$$\begin{align} 2025 &= 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3\\ &= (1+2+3+4+5+6+7+8+9)^2 \end{align}$$

thanks to Nicomachus's theorem.

In other words, 2025 is the square of the 9th triangle number.

But 9 is also the square of a triangle number!

$$\begin{align} 9 &= 1^3 + 2^3\\ &= (1+2)^2 \end{align}$$

Defining some notation for triangle and square numbers,

$$\begin{align} \triangle(n) &:= \frac{n(n+1)}2\\ \square(n) &:= n^2 \end{align}$$

we can summarize things thus:

$$2025 = \square(\triangle(\square(\triangle(2)))).$$

This is fairly benign magic; dabblers in the dark arts may wish to try $\triangle(\square(\triangle(\triangle(2))))$ instead.