This week’s POW asks about dividing a square into squares.

I can partition a square into 4 smaller squares, but I don’t have to make them all the same size. This means I can also partition my square into 6 smaller ones.

What is the largest number of squares you can partition the original area? What is the smallest number of squares for which this isn’t possible? The largest?

Try to prove your answers.

What about cubes?

First, I’m completely confused as to where to even start with this problem. Can’t a square be divided into an infinite number of smaller squares? How can we choose a largest number for which we can partition the original area?

I do think that the smallest number of squares for which this isn’t possible is 2. There is no way for us to divide a square into two squares. This is also true for 3 and 5. However, I am almost positive that it is possible to find a way to divide a square into parts greater than 5. So I guess that makes 5 the greatest number for which this isn’t possible.

How do we prove these? No clue.

If anyone has any ideas that are more helpful with this POW and actually posts them, I will love you with all of my mathy-heart.