50-year old maths problem about an infinite lottery finally solved


Winning an infinite lottery is no easy task

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In an infinite lottery can you create a lottery ticket that always wins? This is the idea behind a 50-year-old maths problem that has now been solved.

In a standard lottery, you have a ticket with a handful of numbers on it and if they match the randomly selected numbers from the lottery, your ticket wins. Each ticket can have several rows in it giving you several chances to win. This means that a long enough ticket could in principle have every possible winning combination meaning that it always wins – although it would cost so much money to do this in reality it wouldn’t be worth it.

But in an infinite lottery things are a little different. The winning collection of numbers is infinitely long, and each ticket can have an infinite number of rows, with each row containing an infinite number of numbers. In this situation, it’s far less obvious whether it’s possible to create a ticket that always wins.

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David Schrittesser and Asger Törnquist at the University of Copenhagen found that the answer is that it isn’t possible to have a ticket that always wins the infinite lottery – half a century after mathematician Adrian R.D. Mathias first posed the question.

“Nobody took the slightest notice for 30 years, and then suddenly people got interested again. It’s very satisfying to see,” says Mattias. Around 20 years ago, some mathematicians rediscovered the problem and started to make progress. Eventually Schrittesser and Törnquist became interested as well, but it took them four years before they cracked it.

The pair used ideas from Ramsey Theory to tackle the problem, a part of mathematics that looks at how order appears in large structure. They found that in an infinite lottery, a sort of structure arises that means the winning numbers clump together, but in a way that means a ticket that always wins just can’t exist.

“With these kinds of problems you don’t sit down and say I’m going to be the one who solves it because everyone has tried,” says Schrittesser. “There’s a little bit of serendipity.”

Journal reference: Proceedings of the National Academy of Sciences, DOI: 10.1073/pnas.1906183116

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