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On this 5 by 5 grid of dots, one square is shown in the diagram. Including this square, how many different sizes of squares can be formed using four dots of this array as vertices?

Feb 22, 2021

#1
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Sadly, I think the best way is to just count.

1 by 1 = 16 squares

2 by 2 = 9 squares

3 by 3 = 4 squares

4 by 4 = 1 square

Note that all the numbers above are squares, this will be a useful pattern when you need to find all the numbers quickly.

1+4+9+16 = 30.

I hope this helped. :))

=^._.^=

Feb 22, 2021
#2
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As a general note, for a square array with $$n$$ dots on each side, the number of squares is:

$$1^2 + 2^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}$$

CubeyThePenguin  Feb 22, 2021
#3
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Ohhh thank you.

I didn't know. :))

=^._.^=

catmg  Feb 22, 2021
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Guest Feb 22, 2021
#5
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The answer is supposted to be 8.

Guest Feb 22, 2021
#8
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Oops, my bad, but it's great that you got the answer.  :))

=^._.^=

catmg  Feb 22, 2021
#6
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The question asks for the number of different sizes.....not the number of squares

2 x2

3x3

4x4

5x5

four different sizes

Feb 22, 2021
#7
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Oh, my bad, but there's also diagonal squares.

=^._.^=

catmg  Feb 22, 2021
#9
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Yes, there are diagonal squares  (hence the total squares 55)

and I suppose these  2 x 2   and 3 x 3  squares ARE a different size

so   SIX different sizes ya think ?

ElectricPavlov  Feb 22, 2021