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Lv 6
? asked in Science & MathematicsMathematics · 6 years ago

Counting squares puzzle?

How many *squares* can be drawn on the 10×10 matrix of dots below so that each vertex coincides with a dot?

[Hint: Squares can be various orientations and sizes.]

Attachment image

1 Answer

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  • Wayne DeguMan
    Lv 7
    6 years ago

    Considering 1 by 1 squares, we have 9 x 9 positions

    i.e. 9 vertically by 9 horizontally

    2 by 2 squares => 8 x 8 positions

    Continuing until the 10 by 10 grid square we get 1

    so, the sum of the first 9 square numbers

    i.e. 1 + 4 + 9 + 16 + ... + 81 = 285

    Now, the diagonal squares are a bit trickier.

    1 by 1 squares 8 x 8 arrangements..i.e. 64

    2 by 2 => 6 x 6 = 36

    3 by 3 => 4 x 4 = 16

    4 by 4 => 2 x 2 = 4

    Bigger ones are not possible

    so, we have 64 + 36 + 16 + 4 => 120

    Then, combining, we have 285 + 120 = 405 arrangements

    An illustrative sketch is below, hope that helps.

    :)>

    Attachment image
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