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## Main Question or Discussion Point

This is not a quiz but I am thinking how to write down a simple math formula to count the total number of squares present in a lattice of NxM points for my 12 year old nephew ? He'll sure be happy if I could turn this into, say, a common sense for pupils like him.

For example,

In a 3x4 lattice there are 20 squares.

I first check 3x3 one (by omitting the last column 3x1) on the right and have

1

then I check 3x3 one (after omitting the first column) on the right to obtain

1

So there are 28 squares. But I have 2 columns overlapped between the two squares I have just checked. And I have no clue how to reason to leave out the overlapped part to acquire the correct result.

For example,

In a 3x4 lattice there are 20 squares.

I first check 3x3 one (by omitting the last column 3x1) on the right and have

1

^{2}+2^{2}+3^{2}= 14 squaresthen I check 3x3 one (after omitting the first column) on the right to obtain

1

^{2}+2^{2}+3^{2}= 14 squaresSo there are 28 squares. But I have 2 columns overlapped between the two squares I have just checked. And I have no clue how to reason to leave out the overlapped part to acquire the correct result.