Every loop hides a square. Find it.
The claim. Draw any loop that does not cross itself. Somewhere on it lie four points that are the corners of a perfect square. This is Toeplitz' conjecture from 1911, also called the inscribed square problem or the square peg problem.
The catch. Nobody has proven it for every possible loop. It is proven for every loop that is smooth, or made of straight pieces, or convex. That covers anything you can draw. The open case is for wild, infinitely wrinkled loops that no pen can make.
Why it must not cross. A loop that goes out along a line and comes back along the same line has all its points on one line, so it holds no square at all. The no-crossing rule is the heart of the theorem, not a technicality.
Counting. For a typical loop the number of squares is odd: one, three, five. The daily loop tells you how many it hides. Find them all.
How to play. Two points on the loop fix the square. In Side mode your two handles are one side, and there are two candidate squares, one on each side. In Diagonal mode the handles are opposite corners and there is one candidate. Guided shows the candidate square while you drag, with lines showing how far the free corners miss the loop: training wheels. Solo shows nothing until you have it.
Made by Romke Jonker. Sibling of Schrödinger's Mouse. All puzzles.