Quizzing the students regularly will increase their competitive spirit and will motivate them to learn 42 times table much faster. And with these almost infinitely improbable numbers, the famous Solutions of the Diophantine Equation (1954) may finally be laid to rest for every value of k from one to 100—even 42. In 2019, Booker, at the University of Bristol, and Sutherland, principal research scientist at MIT, were the first to find the answer to 42. The number has pop culture significance as the fictional answer to “the ultimate question of life, the universe, and everything,” as Douglas Adams famously penned in his novel “The Hitchhiker’s Guide to the Galaxy.” The question that begets 42, at least in the novel, is frustratingly, hilariously unknown.

Professor Booker, who is based at the University of Bristol's School of Mathematics, said: "I feel relieved. In this game it's impossible to be sure that you'll find something. It's a bit like trying to predict earthquakes, in that we have only rough probabilities to go by. If, for example, a room measures 4.42 x 3.96m (14'6" x 13'0") take the nearest measurement above, i.e. 4.50 x 4.00m (14'9" x 13'1") giving you 72 tiles. But slowly, over many years, each value of k was eventually solved for (or proved unsolvable), thanks to sophisticated techniques and modern computers—except the last two, the most difficult of all; 33 and 42. The team also developed ways to efficiently split the algorithm’s search into hundreds of thousands of parallel processing streams. If the algorithm were run on just one computer, it would have taken hundreds of years to find a solution to k=3. By dividing the job into millions of smaller tasks, each independently run on a separate computer, the team could further speed up their search.This involves some fairly advanced number theory, using the structure of what we know about number fields to avoid looking in places we don’t need to look,” Sutherland says. In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x 3+y 3+z 3=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a “Diophantine equation” — a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers. When the sum of cubes equation is framed in this way, for certain values of k, the integer solutions for x, y, and z can grow to enormous numbers. The number space that mathematicians must search across for these numbers is larger still, requiring intricate and massive computations.

There had been some serious doubt in the mathematical and computational communities, because [Mordell’s question] is very hard to test,” Sutherland says. “The numbers get so big so fast. You’re never going to find more than the first few solutions. But what I can say is, having found this one solution, I’m convinced there are infinitely many more out there.” For each computer in the network, they are told, ‘your job is to look for d’s whose prime factor falls within this range, subject to some other conditions,’” Sutherland says. “And we had to figure out how to divide the job up into roughly 4 million tasks that would each take about three hours for a computer to complete.” What do you do after solving the answer to life, the universe, and everything? If you’re mathematicians Drew Sutherland and Andy Booker, you go for the harder problem.

The original problem, set in 1954 at the University of Cambridge, looked for Solutions of the Diophantine Equation x 3+y 3+z 3=k, with k being all the numbers from one to 100. However, solving 42 was another level of complexity. Professor Booker turned to MIT maths professor Andrew Sutherland, a world record breaker with massively parallel computations, and—as if by further cosmic coincidence—secured the services of a planetary computing platform reminiscent of "Deep Thought", the giant machine which gives the answer 42 in Hitchhiker's Guide to the Galaxy. The fact that a third solution to k=3 exists suggests that Heath-Brown’s original conjecture was right and that there are infinitely more solutions beyond this newest one. Heath-Brown also predicts the space between solutions will grow exponentially, along with their searches. For instance, rather than the third solution’s 21-digit values, the fourth solution for x, y, and z will likely involve numbers with a mind-boggling 28 digits.

Booker and Sutherland have now published the solutions for 42 and 3, along with several other numbers greater than 100, this week in the Proceedings of the National Academy of Sciences. With this sleeker version of the equation, the researchers would only need to look for values of d and z that would guarantee finding the ultimate solutions to x, y, and z, for k=3. But still, the space of numbers that they would have to search through would be infinitely large. Multiplication Table is an useful table to remember to help you learn multiplication by 42. You should also practice the examples given because the best way to learn is by doing, not memorizing. Online Practice This was sort of like Mordell throwing down the gauntlet,” says Sutherland. “The interest in solving this question is not so much for the particular solution, but to better understand how hard these equations are to solve. It’s a benchmark against which we can measure ourselves.” The first two solutions for the equation x 3+ y 3+ z 3 = 3 might be obvious to any high school algebra student, where x, y, and z can be either 1, 1, and 1, or 4, 4, and -5. Finding a third solution, however, has stumped expert number theorists for decades, and in 1953 the puzzle prompted pioneering mathematician Louis Mordell to ask the question: Is it even possible to know whether other solutions for 3 exist?Most carpet tiles are produced in squares of 50 x 50cm and supplied in boxes of 12. This table will tell you exactly how many tiles you need for a given room size. Simply read off the length and width and where the columns cross, you'll find the answer. The amount of work you have to do for each new solution grows by a factor of more than 10 million, so the next solution for 3 will need 10 million times 400,000 computers to find, and there’s no guarantee that’s even enough,” Sutherland says. “I don’t know if we’ll ever know the fourth solution. But I do believe it’s out there.”

Beyond the easily found small solutions, the problem soon became intractable as the more interesting answers—if indeed they existed—could not possibly be calculated, so vast were the numbers required. The discovery was a direct answer to Mordell’s question: Yes, it is possible to find the next solution to 3, and what’s more, here is that solution. And perhaps more universally, the solution, involving gigantic, 21-digit numbers that were not possible to sift out until now, suggests that there are more solutions out there, for 3, and other values of k.

As decades went by with no new solutions for 3, many began to believe there were none to be found. But soon after finding the answer to 42, Booker and Sutherland’s method, in a surprisingly short time, turned up the next solution for 3: You can now think of k as a cube root of z, modulo d,” Sutherland explains. “So imagine working in a system of arithmetic where you only care about the remainder modulo d, and we’re trying to compute a cube root of k.” So, the researchers optimized the algorithm by using mathematical “sieving” techniques to dramatically cut down the space of possible solutions for d.