By utilizing finite geometry, mathematicians calculate the minimum number of lottery tickets required for a player to win any prize, not necessarily the jackpot.
It has been joked that lotteries are essentially a form of “tax” aimed at those who can do math. However, there is an undeniable reality that mathematical analysis can indeed help lottery players understand the complex probabilities associated with a game that is always considered a game of chance.
For instance, can we use mathematics to calculate the minimum number of lottery tickets someone could buy to guarantee a win? Mathematicians at the University of Manchester (UK) have discovered that the answer is yes, but this does not mean that this method will make lottery players rich.
There have been various opinions on whether mathematical formulas can somewhat assist lottery players in increasing their odds of winning. (Photo: IFL Science).
According to IFL Science, the research team examined the Lotto lottery system of the national lottery program in the UK, where players choose six numbers from 1 to 59. The goal for players is to be lucky enough to match the numbers drawn by the lottery company. A player can win a prize by matching a pair of numbers to the jackpot (matching all six numbers), which is currently valued at £7.8 million ($9.9 million).
Accordingly, Dr. David Stewart and Dr. David Cushing, two members of the research team, found a way to match at least two numbers out of all 45,057,474 possible combinations. It turns out that all a player needs to do is purchase a minimum of 27 lottery tickets.
To describe the possible combinations, the research team employed a process called finite geometry, where different numerical combinations are represented as points in a geometric shape.
Here, they used the “Fano plane” – a geometric structure where pairs of numbers are plotted on or within triangles, and straight lines or circles connect them. Each line passes through three pairs, creating one of the sets of six numbers with a high probability of winning the jackpot.
Mathematically, the allure of the calculations is not about winning the lottery and making a lot of money. Instead, the research team aimed to determine the minimum number of tickets a player could buy to win some kind of prize, even if it is not substantial.
By using finite geometry, mathematicians calculated that 27 tickets, each with a different set of numbers, is sufficient for a player to win at least one prize, though not necessarily the jackpot. (Photo: University of Manchester).
Mathematics Helps Win, but Profit is Uncertain
After calculations, the number of 26 tickets is considered mathematically very ‘difficult’ for someone to win a prize. In contrast, 27 is the minimum number of tickets required, with each ticket needing to have a different number combination. This is viewed as the secret for players to at least win some prize (not necessarily the jackpot).
However, at a cost of £2 per ticket, totaling £54 for 27 tickets, players are unlikely to profit even if they win.
In the UK, a player wins a cash prize of £30 if their ticket has 3 matching numbers with those drawn. However, players can receive a free Lucky Dip ticket if they correctly guess two numbers. The Lucky Dip ticket offers players another chance at a ‘life-changing’ win, except that the numbers are chosen randomly for them.
According to mathematician Peter Rowlett’s estimates, in 99% of cases, the investment will not yield a proportional return. You might win, but not by much. Clearly, the theory can only support you to a certain extent.
In reality, the research team itself tried to apply their method practically. They purchased 27 lottery tickets before the drawing on July 1, 2023. As a result, they had 3 tickets matching at least 2 numbers, earning them 3 free Lucky Dip tickets. However, none of these tickets won any prizes. Thus, it resembles a more intriguing mathematical method rather than a quick money-making scheme.
Interestingly, the mathematical method developed by the University of Manchester team has also attracted the attention of lottery companies. A spokesperson for Camelot, the company that operates the national lottery in the UK, affirmed that this research provides many interesting perspectives.
“Our approach has always been to attract as many players as possible to spend as little money as possible on the lottery,” said the Camelot representative.
“It is important to remember that, ultimately, a lottery is a lottery. Like all other games based on the national lottery program, all winning Lotto numbers are drawn randomly. Any number has an equal chance of being drawn. Therefore, every combination entered in a draw has an equal chance of winning.”
