The card game Set is as simple as ABC. Yet the maths behind it have puzzled mathematicians for decades. Delft Mathematician Dr. Dion Gijswijt helped unravel some of its mysteries.
“Set is a simple game, popular with children,” Gijswijt said. “Its goal is simple: to find special triples, called ‘sets,’ within a deck of 81 cards.” The maths behind it have confused researchers for years, however. Questions such as “How big is the largest collection of cards that contains no set?” are the kind of questions mathematicians deal with in the field of combinatorics.
The answer to the above question is 20. Set, however, is still quite simple. Each card displays a different design with four attributes. They differ in colour (cards can be red, purple or green), shape (oval, diamond or squiggle), shading (solid, striped or outlined) and number (one, two or three copies of the shape).
In theory, one could extend the designs with more than four attributes. “How big would then be the largest collection of cards that contains no set? In extremal combinatorics, this is known as the cap set problem,” Gijswijt said. Together with an American colleague Gijswijt developed a technique, a so-called polynomial method, to calculate this. The simplicity of the solution has stunned mathematicians.
The polynomial method has many implications. It is useful in linear algebra, for instance. One can use the method to multiply huge matrices quickly. The paper by Gijswijt and Ellenberg was published on May 30th on ArXiv.
Within only a few weeks’ time, it elicited the publication of five other papers. It seems a train is set in motion. Mathematicians around the world are publishing about this new finding with the speed of light, mostly on ArXiv, a repository of electronic preprints.
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