Team solves 50-year-old lottery ticket math mystery

After years of work, mathematics researchers have answered a mysterious half-century-old riddle.

Is there a lottery ticket that always wins? That’s the popular version of a theoretical conundrum posed in 1969 by English mathematician Adrian RD Mathias within the field of set theory, an area dealing with infinity in mathematics.

The problem remained a mystery throughout the 70’s, 80’s, and 90’s, as set theorists the world over tried their best to solve it. Asger Dag Törnquist, an associate professor of the University of Copenhagen’s mathematical sciences department, learned about the problem in 2002 while completing his doctoral dissertation at the University of California, Los Angeles.

“Research in the area had gone dormant from the 1990’s on because no one was making any progress towards a solution. I was fascinated because it was an old problem that dealt with our understanding of infinity in mathematics. Even then, it became a dream of mine to solve the mystery, even though I had no idea of how to accomplish what had been elusive for others over decades,” he says.

A MAD mystery

Mathias researched order and structure, things that occur spontaneously in sufficiently large mathematical systems. Today, this is known as Ramsey Theory, named after British mathematician and philosopher Frank Ramsey. Mathias’ research pointed out that there was a profound correlation between Ramsey Theory and what he called MAD families, but he was unable to prove the existence of such a relationship.

“A MAD family can be thought of as a kind of lottery ticket that always wins in a peculiar, infinite lottery game. In this game, lottery tickets have an infinite number of rows of whole numbers, and each row itself has infinitely many numbers. And, a ticket may have so many rows that they simply cannot be numbered,” says Törnquist.

Mathias asked the math world if the order and structure that we know are there, as per Ramsey Theory results, prevent the existence of a MAD family, i.e., a ticket that always wins.

Winning the math lottery

“In 2014, I decided to rethink the problem from scratch and found a whole new way of tackling it. Alongside the original mystery, Mathias had formulated a sort of baby-version of the mystery. Neither had been solved. I managed to solve the baby version of mystery, which I then wrote an article about,” explains Törnquist, who approached the solution with postdoctoral researcher David Schrittesser.

As a result, many mathematicians from around the world reacted. The article suddenly reignited research in the area. Researchers in other parts of the world began to build upon the researchers’ article and more and more pieces of the puzzle began falling into place.

“We were in the midst of writing an article meant to address yet another small piece of the puzzle, when we realized that we may have been closer to solving the entire riddle than we had believed. From then on, things moved quickly. A few weeks later, we had the solution,” he recounts.

After five years of work, Törnquist and Schrittesser discovered that complete coincidence does not exist.

“We found out that lottery ticket numbers clump up in such a way that there is no certainty of a winner, which was what Mathias had guessed would happen, but had been unable to prove. This confirms that one cannot assemble such a type of a lottery ticket without the emergence of certain patterns and regularities in ticket numbers. As such, there is no lottery ticket that always wins Mathias’ lottery game,” concludes Törnquist.

The research appears in the Proceedings of the National Academy of Sciences.

Source: University of Copenhagen

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