I loved the comments on yesterday’s post. Anton’s solution to the puzzle was very creative. My only caveat is that it might be hard for someone looking at that shape to realize that it is suggesting a puzzle about squares, since the shape isn’t built from squares, but rather from shapes with a width:height ratio of 4:3.
My own solution is based on the observation that you can always fold up a corner of a shape made up of squares, thereby removing a square while preserving the length of the perimeter:
Applying this idea iteratively to the puzzle of finding a shape with the same perimeter as a square, but half the area of that square, I found the following solution:
Although to make things more fun (and prettier) in posing the puzzle, I would present the shape rotated 45o, so it would look more like this:
The puzzle Alec discussed in his comment — to find a shape with the same perimeter as a circle, but half the area of that circle — has a solution very different from the one he proposes, reminiscent of my recent Yin Yang post. Actually, it has an infinite number of solutions, of which these are the first two in a series:
The yin-yang solutions are great. It’s much easier to tell that they are indeed solutions.