Harvard-MIT Mathematics Tournament

Solution:
The last digit of a square is never 7. No two-digit squares begin with 7. There are no 3-digit squares beginning with the digits $17, 27, 37$, or $47$. In fact, the smallest square containing the digit 7 is $576 = 24^{2}$. Checking the next few numbers, we see that $25^{2} = 625$, $26^{2} = 676$, $27^{2} = 729$, $28^{2} = 784$, and $29^{2} = 841$, so the answer is 27.