Squaresville

How many squares are there in a 3 x 3 grid like this one?

Don’t answer so fast! Sure, there are 9 small 1 x 1 squares, but there are 2 x 2 squares, too. And the whole grid is a 3 x 3 square. So what’s your answer now?

 Count the squares in these figures:

 What size grid would give you exactly 100 squares? Why isn’t it a 10 x 10 grid?

Solution:

There are 9 1x1 squares, but 4 2x2 squares and 1 3x3 squares, so there are 14 squares in the figure.

a) I made a table to count the squares in this 4x4 grid:

b) Here's a table to help count the squares in this 4x5 grid. I flipped the rows and columns and started to see patterns!

c) I noticed the descending factors in the table, and they continued with the 6x4 grid:

That must mean a 7x4 grid would have 60 squares, and so on. We're looking for 100:

So an 11x4 grid would contain exactly 100 squares.

I was looking for other grids with 100 squares so I extended the pattern to the 5's:

    The totals seemed to be going up by 15, meaning the next one, for a 8x5 grid, would be 100 squares!

Hmm. I wondered if the next smaller odd number by the next higher integer would continue the pattern, so I wrote a program in Sage that would give the total number of squares in an m x n grid (n being the smaller number):

 but I found that a 7x6 grid has 112 squares. Similarly, a 13x3 grid has 74 squares. But now I could easily calculate the number of squares in any grid I wanted! The only other one I found that came to exactly 100 was the obvious 100x1 grid.

Once you have the total for an m x m grid, however, the next total (m x (m+1)) goes up by . We saw the 4x4 grid had 30, then the 4x5 grid was 10 higher, then the 6x4 grid was 10 higher, and so on. The 5-grids went up by 5(6)/2 = 15, the 6 grids went up by 6(7)/2 = 21 and so on.

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