[{"id":310692,"date":"2026-09-03T06:00:34","date_gmt":"2026-09-03T10:00:34","guid":{"rendered":"https:\/\/3quarksdaily.com\/?p=310692"},"modified":"2026-09-01T02:52:08","modified_gmt":"2026-09-01T06:52:08","slug":"puzzles-for-fun-and-profit","status":"publish","type":"post","link":"https:\/\/3quarksdaily.com\/3quarksdaily\/2026\/09\/puzzles-for-fun-and-profit.html","title":{"rendered":"Puzzles for Fun and Profit"},"content":{"rendered":"

by Jonathan Kujawa<\/strong><\/p>\n

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Gary Musser<\/figcaption><\/figure>\n

Lots of people enjoy a good puzzle. Take Sudoku puzzles<\/a>. \u00a0Like many good puzzles, the rules of Sudoku are simple. The challenge is to figure out the implications of each possible move and to use those implications to reason our way to the solution. You probably know how this goes: you try to put a 5 in one box, but that leads to problems down the road when that choice forces you into a situation that violates the rules. The only way out is to relent and admit that 5 must go into a different box.<\/p>\n

At its heart, mathematics is a vast Sudoku puzzle. Through reasoning, clever observations, and following the implications wherever they go, we learn new things about the mathematical universe. For example, if xn<\/sup> + yn<\/sup> = zn<\/sup> has a solution where n, x, y, z are integers and n > 2, then certain geometric shapes called elliptic curves must have a unique associated function with special properties that make it a “modular form” [1]. Hard work by Andrew Wiles and many others showed that the relevant elliptic curves cannot be associated to a modular form; hence, we must relent and admit that Fermat’s equation has no solutions.<\/p>\n

Mathematicians love nothing better than solving a math problem by conjuring new knowledge out of thin air using nothing but creative reasoning. That creativity and use of the human mind are sadly missing from much of our math education.<\/p>\n

The problems of puzzles are very near the problems of life.<\/em>\u00a0\u2014Erno Rubik<\/p>\n

I was recently reminded of the puzzle-like nature of mathematics. In conjunction with the annual Teachers of Teachers of Math conference hosted by my university, my department hosted a celebration of the life and career of our colleague, Gary Musser. Gary was a long-time faculty member in our department who, sadly, passed away this year. I had the pleasure of chatting with Gary and his wife, Irene, when I first arrived at Oregon State. Although Gary retired several decades ago, he and Irene continued to have a deep interest in the department, in mathematics, and in teaching mathematics. In their honor, the department gives the Gary Musser Award each year to support outstanding prospective elementary or middle school teachers.<\/p>\n

Why an award for prospective teachers? Well, while Gary’s PhD thesis was on a highly theoretical and advanced area of mathematics, his professional energies were focused on an even harder problem: how to train school teachers to be effective at teaching children mathematics. This problem is doubly hard. After all, young children need a good educational experience in math, and at the same time, the teachers of young children are often not “math people” themselves. Gary understood the importance of solving this problem. He spent his career having a profound effect on how mathematics is taught in Oregon and across the United States.<\/p>\n

One of Gary’s strongest-held beliefs was in the importance of puzzles. Specifically, creative reasoning puzzles that tied into mathematics. By connecting teachers and, by extension, their students with the puzzling nature of mathematics, Gary was convinced he could unlock an understanding and a love of the subject. Gary, Blake Peterson, and William Burger wrote a seminal textbook entitled Mathematics for Elementary Teachers<\/em><\/a>. Blake told us that Gary insisted that puzzles be at the forefront in their text. Gary was a wizard at finding (or inventing) puzzles that caught the imagination while also revealing an important pedagogical point. During the memorial, Blake and his co-presenter, Mike Shaughnessy, shared the following puzzle:<\/p>\n

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What fraction of the rectangle is contained in the orange triangle?<\/figcaption><\/figure>\n

You might think there is not enough information to solve the puzzle. I promise there is. You might try to calculate using the Pythagorean theorem, the formulas for the areas of rectangles and triangles, and various other geometric machinations. But those fancy tools are unnecessary. With a single insight, the puzzle almost solves itself. Here’s a hint \u00a0[2].<\/p>\n

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What fraction of the square is purple?<\/figcaption><\/figure>\n

Blake and Mike then shared a puzzle which appears even harder at first glance, but likewise yields to a bit of cleverness. In the picture on the right, what proportion of the big square is contained in the purple squares? On this problem, the mathematicians are at a disadvantage. If you have taken calculus, you will be tempted to turn this problem into an infinite series and use high-powered techniques to compute that series. The secret is to be patient, think deeply, and look for an insight that cuts straight through to the solution [3].<\/p>\n

There is another type of puzzle that is devilish fun. Again, it seems like you haven’t been given enough information, and yet it is possible to answer the question with careful thought. They are sometimes called I Don’t Know puzzles because they typically involve a scenario where the characters say “I don’t know” some number of times, thereby revealing information that eventually leads to the solution. Here’s one:<\/p>\n

Irene and Gary are told that integers \ud835\udc65 and \ud835\udc66 have been chosen such that 1 < \ud835\udc65 < \ud835\udc66 and \ud835\udc65 + \ud835\udc66 < 100. Irene is given the value \ud835\udc65 + \ud835\udc66, and Gary is given the value \ud835\udc65 \u22c5 \ud835\udc66. They then have the following conversation.<\/p>\n