Square Palindromes

A palindrome reads the same forwards and backwards. Some perfect squares are palindromes: $$1^2 = 1, \qquad 11^2 = 121, \qquad 111^2 = 12321.$$ Why does this pattern hold, how far does it run, and what other palindromic squares are there?

Solution

The pattern belongs to the repunits, the numbers made only of ones. As long as no column total reaches ten, squaring a string of $k$ ones makes the digits climb $1, 2, \dots, k$ and back down: $$1111^2 = 1234321, \qquad 11111^2 = 123454321,$$ and so on up to $$111111111^2 = 12345678987654321 \quad (\text{nine ones}).$$ The reason is that a string of $k$ ones squared is the sum of $k$ shifted copies of that string, and the $j$th column simply counts how many copies overlap there, a count that rises to $k$ in the middle and falls away symmetrically. With ten ones or more, a column total reaches ten, carrying begins, and the neat palindrome is lost: $1111111111^2 = 1234567900987654321$, which is not a palindrome.

Palindromic squares need not come from palindromic roots. Among the smaller ones, $$\begin{gathered} 22^2 = 484, \quad 26^2 = 676, \quad 212^2 = 44944,\\ 264^2 = 69696, \quad 836^2 = 698896, \end{gathered}$$ and of these roots only $22$ and $212$ are themselves palindromes.