For what number of nonoverlapping unit squares can a figure be formed whose perimeter is numerically equal to the area?
Adding one square offset by 1/2 unit adds 1 unit area and 3 units perimeter so it subtracts 2 units from the quantity (AP).
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 nonoverlapping unit squares is:
16, 18, and all integers even and odd greater than or equal to 20

Posted by Larry
on 20140710 10:57:00 