The Shape of Problem-Solving

Take any number. Reverse its digits. Add them together. Keep going until you hit a palindrome.

Most numbers get there eventually. Some, like 196, might never arrive. We've tested it billions of times. Still no palindrome, which makes it a lychrel candidate.

This visualization plots 10,000 numbers. You'll notice horizontal bands where clusters of numbers all resolve at the same step. And gaps between them. The red diamonds at the bottom are numbers that haven't found their palindrome yet.

It starts to look like the shape of problem-solving. Easy problems cluster together. Hard ones float alone. And sometimes you're just in the gap, waiting for the next breakthrough. You can see those breakthroughs visualized by many numbers converging to the same palindrome, creating these plateaus.

Interact: Click and drag to rotate. Scroll to zoom. Hover for details.

Note: This visualization uses starting numbers 10–10,000 with a cap of 500 iterations. Some numbers beyond this range require thousands of iterations to converge. The patterns hold, but the gaps get wider.