Posted by **mathemagiacian** on Thursday, August 15, 2013 at 6:19pm.

A four by four grid of unit squares contains squares of various sizes (1 by 1 through 4 by 4), each of which are formed entirely from squares in the grid. In each of the 16 unit squares, write the number of squares that contain it. For instance, the middle numbers in the top row are 6s because they are each contained in one 1*1 square, two 2*2, two 3*3, and of course the one 4*4.

What is the sum of all sixteen numbers written in this grid?

How about a ten by ten grid?

I really hope to see the answer. Thanks! :)

- MATH -
**MathMate**, Thursday, August 15, 2013 at 7:34pm
What did you get?

You would be well to do the 4x4.

If you have difficulties, start with a 1 by 1, then 2 by 2 and then 3 by 3 grid. You will find a pattern that you can use later.

Once done, do an n × n grid and check if the answer for n × n works for n=4.

- MATH -
**HI**, Tuesday, August 12, 2014 at 8:10pm
104

- MATH -
**yea**, Saturday, November 15, 2014 at 5:48pm
The Grid looks like this:

4 6 6 4

6 10 10 6

6 10 10 6

4 6 6 4

- MATH -
**哈羅**, Wednesday, August 12, 2015 at 2:23am
what about ten by ten

- MATH -
**AoPS Solution**, Thursday, September 3, 2015 at 10:02am
We can use the same logic-based approach as the first part, but this gets difficult for large grids, so we resort to imagination to help us out.

Instead of looking at each unit square and wondering how many squares (unit or larger) it is contained in, we turn the thinking around: We consider every possible square (unit or larger) and think about how many unit squares are in it.

Suppose we mark the grid as follows: For every 1x1 square (and there are 10^2 = 100 of them), we mark each small square in the 1x1 square. We end up with a mark in all 100 squares. We then do this for all 2x2 squares (there are 9^2 = 81 of them), marking each of the four small unit squares. We do this for every size of square, up to the 10x10 square.

Then for each small square, the number of marks in each square is equal to the number of squares the small square is contained in.

There are 100 1x1 squares, so we made 100*1=100 marks. There are 81 2x2 squares, so we added another 81*4 marks, and so on. Therefore, the total number of marks is 100 * 1 + 81 * 4 + 64 * 9 + 49 * 16 + 36 * 25 + 25 * 36 + 16 * 49 + 9 * 64 + 4 * 81 + 1 * 100 = 5368

Answer directly from AoPS

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