How much lottery do you need to buy to ensure you win something in the UK National Lottery? Mathematicians at the University of Manchester have answered this question by focusing on the flagship “Lotto” game, which draws six random numbers from 1 to 59, Dr David Stewart and Dr David Cushing comfort that 27 is the lowest possible number. of tickets needed to guarantee a prize, though, more importantly, with no guarantee of making the initial spend profitable.
They describe the solution using a mathematical system called finite geometry, which is centered around a triangle-like structure called the Fano plane. Each point of the structure is drawn with pairs of numbers and connected with lines: each line generates a set of six numbers, which is equal to one ticket.
What is the method?
Three Fano planes and two triangles are needed to cover all 59 numbers and generate 27 sets of tickets. Choosing tickets in this way ensures that no matter which of the 45,057,474 possible draws to prevent, at least one of the tickets will have at least two numbers in common. Of any draw of six, two numbers must appear in one of the five geometric structures, ensuring that they appeared on at least one ticket. But Dr. Stewart and Dr. Cushing say the hard work is actually showing that you can’t get the same result with 26 tickets.
Six numbers that do not appear on any ticket together
Dr David Stewart, a professor of pure mathematics at the University of Manchester, said in a statement: “Fundamentally, there is a stress that comes from the fact that there are only 156 entries in 26 tickets. This means that many numbers cannot appear one time”. many times. Eventually you will see that you will be able to find six numbers that do not appear on any ticket together. In graph theory terms, we ended up proving the existence of an independent set of size six.”
It is profitable?
Although a win is guaranteed, the researchers say the chances of winning are highly unlikely and should not be used as a reason to gamble. The 27 lottery tickets would cost you £54. And Peter Rowlett, a mathematician at The Aperiodical website, has shown that in almost 99% of cases, he would not get that money back.
By testing the theory in the lottery draw on July 1, 2023; the researchers hit just two balls on three of the tickets, the reward being three lucky guesses in a subsequent lottery, each resulting in nothing.
The researchers say the finding is computationally interesting. They use a fifty-year-old programming language called Prolog, which they say makes it one of the oldest examples of real artificial intelligence.
