# The ever-winning lottery ticket: Mathematicians solve a dusty mystery

September 09, 2019Is there a lottery ticket that always wins? So goes the popular version of a theoretical conundrum posed in 1969 by English mathematician Adrian R.D. 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. Associate Professor Asger Dag Törnquist of the University of Copenhagen's Department of Mathematics was introduced to the problem in 2002 while completing his doctoral dissertation at the University of California, Los Angeles (UCLA).

"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.

**MAD families**

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 is 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.

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

**The 'baby-mystery' proved decisive**

Asger Dag Törnquist's shouldered his dream of solving Mathias' question for several years abroad until he began working at the University of Copenhagen's Department of Mathematical Sciences in 2011. This marked the beginning of a period during which Törnquist and David Schrittesser, his Austrian postdoctoral researcher, would gradually approach the solution.

"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.

As a result, a great 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 UCPH 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," recounts the mathematician.

**Solution: An ever-winning lottery ticket does not exist**

After five years of work, Asger Dag Törnquist and David Schrittesser had their research article on Adrian Mathias' "lottery ticket" accepted to the prestigious American scientific journal, The

*Proceedings of the National Academy of Sciences*(

*PNAS*). The two researchers 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 Asger Dag Törnquist.

-end-

University of Copenhagen

**Related Mathematics Articles:**

A new method for boosting the learning of mathematics

How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

Could mathematics help to better treat cancer?

Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

People can see beauty in complex mathematics, study shows

Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Improving geothermal HVAC systems with mathematics

Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

How the power of mathematics can help assess lung function

Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Mathematics pushes innovation in 4-D printing

New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

## Trending Science News

**Current Coronavirus (COVID-19) News**

## Top Science Podcasts

We have hand picked the**top science podcasts of 2020**.

**Now Playing: TED Radio Hour**

**Uncharted**

There's so much we've yet to explorefrom outer space to the deep ocean to our own brains. This hour, Manoush goes on a journey through those uncharted places, led by TED Science Curator David Biello.

**Now Playing: Science for the People**

**#556 The Power of Friendship**

It's 2020 and times are tough. Maybe some of us are learning about social distancing the hard way. Maybe we just are all a little anxious. No matter what, we could probably use a friend. But what is a friend, exactly? And why do we need them so much? This week host Bethany Brookshire speaks with Lydia Denworth, author of the new book "Friendship: The Evolution, Biology, and Extraordinary Power of Life's Fundamental Bond". This episode is hosted by Bethany Brookshire, science writer from Science News.

**Now Playing: Radiolab**

**Dispatch 1: Numbers**

In a recent Radiolab group huddle, with coronavirus unraveling around us, the team found themselves grappling with all the numbers connected to COVID-19. Our new found 6 foot bubbles of personal space. Three percent mortality rate (or 1, or 2, or 4). 7,000 cases (now, much much more). So in the wake of that meeting, we reflect on the onslaught of numbers - what they reveal, and what they hide. Support Radiolab today at Radiolab.org/donate.