Write \sigma(m) for the sum of the decimal digits of m. Among all integers n with 1 < n < 100, how many satisfy \sigma(n^2) = n, and what is the smallest of them?

Exactly one such n, namely n = 9, since 9^2 = 81 and 8 + 1 = 9.

Every number is congruent to its own digit sum modulo 9, so \sigma(n^2) = n forces n \equiv n^2 \pmod 9.

The Square Whose Digits Add Back to the Root

Issue 27 Warm-up Published 14 Oct 2026 By The Chalkboard Desk

The problem

Write for the sum of the decimal digits of . Among all integers with , how many satisfy , and what is the smallest of them?

Take a minute

Reveal hint 1

Every number is congruent to its own digit sum modulo , so forces .

Reveal hint 2

That congruence leaves only seven candidates below , once you also notice that has at most four digits and therefore .

AnswerExactly one such , namely , since and .

Derivation