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My solution

My solution is as follows:

If I look at the entire ten box rectangle, there can only be one ten box rectangle.

But if I look at nine box rectangles there are two possibilities: 1-9 and 2-10.

If I look at 8 box rectangles there are three possibilities, 1-8, 2-9, 3-10.

I have worked out that each I decrease the number of boxes by one the number of possibilities increases by one.

The total number of possibilities is the sum of the numbers 1-10 because its 1+2+3+4+5+6+7+8+9+10.

The sum is 55

An easy way to figure out the sum of a sequence of consecutive numbers is to add the first and last numbers and multiply the result by the number of pairs of numbers. Here this would be 1 plus 10 = 11 x5 (pairs) = 55