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Union of squares (Posted on 2014-07-09) Difficulty: 3 of 5
For what number of non-overlapping unit squares can a figure be formed whose perimeter is numerically equal to the area?

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Solution set of numbers | Comment 9 of 10 |

Adding one square offset by 1/2 unit adds 1 unit area and 3 units perimeter so it subtracts 2 units from the quantity (A-P).
Adding on a unit square like this to a 4x5 rectangle makes A=P=21; and doing so to a 3x8 rectangle makes A=P=25.
Also, adding any number of 2x1 rectangles so that the side of length 2 is fully touching the wall of any existing structure (with length 1 sides not touching) adds 2 to both A and P.
I haven't ruled out smaller odd structures, but so far the allowable number of non-overlapping unit squares is:


16, 18, and all integers even and odd greater than or equal to 20


  Posted by Larry on 2014-07-10 10:57:00
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