TITLE: When written in decimal notation, every square number has at most 1000 digits that are not 0 or 1. True or false?
QUESTION [6 upvotes]: When written in decimal notation, every square number has at most $1000$ digits that
  are not $0$ or $1$. True or false? 

This question is from an admissions quiz, so no calculators should be used.

REPLY [7 votes]: $$(10^n+n)^2=10^{2n}+2n \cdot 10^n+n^2$$
So each even number can appear as pattern in a square number. If one chooses $n=5m$ then $2n=10m$ and so each $m \in \mathbb{N}$ appears as digit pattern in the above sequence.