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Math Puzzle: Go to great lengths
By Heinrich Hemme · 2026-06-16 · via Scientific American

A square is divided into eight rectangles of equal area. One of these rectangles has a width of 8. What are the lengths of the sides of the square?

Graphic shows a light yellow square divided into eight rectangles that have varying dimensions but equal areas. One of the rectangles is highlighted in a darker yellow and one of its shorter sides is labeled with the number eight.

If the rectangle with width 8 has length a, its area is A = 8a. Now the dimensions of the other rectangles can be determined successively. We have B = C = a/2 × 16, D = a/3 × 24, E = 4a/3 × 6, F = 4a/15 × 30, G = 8a/5 × 5, and H = 8a/35 × 35. The square therefore has a side length of 35.

The same figure above is shown with labels added for the area (capital A) and longer side (lowercase italic a) of the darker yellow rectangle. The other seven rectangles are labeled with capital letters B, C, D, E, F, G and H.

We’d love to hear from you! E-mail us at games@sciam.com to share your experience.

This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.