Sondra J.

asked • 09/10/14

how many squares of any size are in a 5 x 5 grid?

directions say to make it a smaller problem to solve the larger problem.

1 Expert Answer

By:

Matt H.

Actually, since it says "squares of any size," you have several more than the 25, because within the 5 x 5 grid, you will see 4 x 4 squares, 3 x 3 squares, and 2 x 2 squares, in different positions throughout the grid, plus the big 1 x 1 square around the outside...
 
So you need to draw a square, divide it into a 5 x 5 grid, and then spot all the configurations of 4 x 4, 3 x 3, 2 x 2, etc. and add them to the 25 "obvious" ones and the 1 big one...
 
hope this helps!
 
Matt
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09/11/14

Stephen K.

tutor
Matt,
 
Given your interpretation of the problem, and I think you are correct, I only see one additional solution and that is the 1 large square to fill the entire 5 x 5 grid making a total of 26.  Since 25 is only divisible by 5 and 1 I don't see any other combination that would "fill" the gird.  Do you see another possible solution?  Thanks for your input.
 
SK
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09/11/14

Stephen K.

tutor
I am stipulating here that all squares must be the same size, if we allow combinations of different size square, its still doable, just gets a bit messy.
 
SK
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09/11/14

Matt H.

Hi Stephen, thanks for your posts!
 
I'm interpreting the direction to mean that you can count any size square, whether it's 5 x 5, 4 x 4, etc., all the way out to the giant one.
 
If you allow for every size, AND you allow them to overlap, you'll get ...51 squares! :-)
 
I wish I could draw it out, but maybe try this: draw a 5 x 5 grid, and number the boxes 1-25, starting in the upper left and going left to right (so your second row starts with 6, etc.)
 
Now, notice you can make a 2 x 2 square out of boxes 1,2,6, and 7. Then you can overlap another one made out of 2,3,7, and 8.
 
If you follow this general idea and overlap at every possible place, you get a lot of boxes!
 
I came up with 51, but I may have miscounted.
 
Whatcha think?
 
THANKS,
 
Matt
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09/11/14

Stephen K.

tutor
You may be right, I played with it after I saw your note, didn't try to work overlapping squares, but made an assumption that all squares did not have to be the same size, in which case you get solutions such as 1-4x4 + 9-1x1, or 1-3x3 + 3-2x2+ 4-1x1, etc. and I came up with 11 combinations of that sort.  I'm not really sure what the "correct" approach actually is based on the way the problem is stated, it may well be a combination of "all-of-the-above".  
 
Thanks for the input though, it was an interesting exercise.
 
Steve
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09/11/14

Avi B.

55 It follows the rule of the sum of 1^2+2^2+3^2.....
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01/20/20

Sunday B.

The answer is 55: 1x1 squares =25 2x2=16 3x3=9 4x4=4 5x5=1 25 + 16 + 9 +4 +1=55 total possible squares
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10/28/20

Anivia M.

The answer is 225
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03/12/21

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