MHB Can squares of any size fit perfectly into a 16.5cm x 14cm rectangle?

[TPceebee]

Hi, I'm a product designer looking for some help.

I need to fit squares into the dimensions of a rectangle. There can be any amount of squares just as long as the squares are complete and their dimensions are complete decimals.

The dimensions are 16.5cm x 14cm. I don't want the squares to be any larger than 2.5cm or any smaller than 0.5cm.

 

[MarkFL]

I would convert the dimensions of the rectangle to mm, so that you have 165 mm X 140 mm. Next, let's look at the prime factorization of both measures:

$$140=2^2\cdot5\cdot7$$

$$165=3\cdot5\cdot11$$

We see that the only common factor is 5, so your only choice (for complete tessellation) is to tile the rectangle with squares 5 mm X 5 mm, or 0.5 cm X 0.5 cm.