# Tiling squares

This week’s Fiddler is about tiling a square with smaller squares.

Suppose you have infinitely many 3-by-3 cm tiles and infinitely many 5-by-5 cm tiles. You want to use some of these tiles to precisely cover a square whose side length is a whole number of centimeters. Tiles may not overlap, and they must completely cover the larger square, without jutting beyond its borders. What is the smallest side length this larger square can have, such that it can be precisely covered using at least one 3-by-3 tile and at least one 5-by-5 tile?

Extra credit:
This time, you have an infinite supply of square tiles for each odd whole number side length (as measured in centimeters) greater than 1 cm. In other words, you have infinitely many 3-by-3 cm tiles, infinitely many 5-by-5 cm tiles, infinitely many 7-by-7 cm tiles, and so on. You want to use one or more of these tiles to precisely cover a square whose side length is $N$ cm, where $N$ is an integer. Once again, tiles may not overlap, and they must completely cover the larger square without jutting beyond its borders. What is the largest integer N for which this task is not possible?

My solution:
[Show Solution]