Manchester University discover minimum number of lottery tickets needed to win

The academics used the flagship game 'Lotto' which draws six random numbers from 1 to 59. Credit: PA Images

The minimum number of lottery tickets a person needs to buy to guarantee a win has been discovered by researchers.

Mathematicians found 27 tickets is the lowest number needed to promise a win - but despite this, it still does not mean you will make a profit.

Using a complex maths-based system they were able to show that no matter which of the 45,057,474 possible draws occurs, at least one of the tickets will have at least two numbers in common.

The academics at The University of Manchester used the National Lottery's flagship game 'Lotto' which draws six random numbers from one to 59.

They describe the solution using a mathematical system called finite geometry, which centres around a triangle-like structure called a Fano plane.

The discovery was made using what's called a Fano plane. Credit: The University of Manchester

Each point of the structure is plotted with pairs of numbers and connected with lines – each line generates a set of six numbers, which equates to one ticket. 

It takes three Fano planes and two triangles to cover all 59 numbers and generate 27 sets of tickets. 

From any draw of six, two numbers must appear on one of the five geometric structures, which ensures they appear on at least one ticket.  

Mathematicians at Manchester University have made the discovery. Credit: PA Images

Dr David Stewart, a Reader in Pure Mathematics at The University of Manchester, said: “Fundamentally there is a tension which comes from the fact that there are only 156 entries on 26 tickets.

"This means a lot of numbers can’t appear a lot of times. Eventually you see that you’ll be able to find six numbers that don’t appear on any ticket together.

"In graph theory terms, we end up proving the existence of an independent set of size six.” 

Although guaranteed a win, the researchers say that the chances of making a profit are very unlikely and shouldn’t be used as a reason to gamble.  

The academics say that 27 tickets does not guarantee a profit. Credit: PA Images

The 27 lottery tickets would set you back £54.

Peter Rowlett, a mathematician from The Aperiodical website, has shown that in almost 99% of cases, you would not make that money back.  

When putting the theory to the test in the lottery draw on 1 July, the researchers matched just two balls on three of the tickets, the reward being three lucky dip tries on a subsequent lottery, each of which came to nothing. 

The researchers say the finding is interesting from a computational point of view.

They use a 50-year-old programming language called Prolog, which they say makes it one of the oldest examples of real artificial intelligence. 

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