September Problem of the Month

Shankar’s Squares

Shankar’s fond of squares.  So fond, he has every size of square you could possibly need, and for each size, he has as many as you would ever require.  He also likes to put squares together to make larger squares.  Since you seem pretty smart, he asks you the following question:

  • For which numbers n can you make 1 square by piecing together n squares?  (The n squares are allowed to be all of the same size, or all different sizes, or any combination of sizes you like.)

For example, you can make 1 square from 4 squares by using 4 squares all of the same size and putting them together in the obvious way.

Shankar’s asking a lot of people, and he’ll give a small prize to the person with the best answer.  At the minimum, your answer should be a complete and simple description of all n for which this can be accomplished.  To make your answer better than anyone else’s though, you should give a clear explanation of why your answer is correct.

[See the rules in the PHSC 2nd floor display case for more information.]