And Then, There Were None (Under 100)

There was big news in the world of maths last week, when Andrew Booker announced that he had solved a long-standing puzzle in maths on Reddit in a post titled Life, the Universe, and Everything, which simply linked to an equation showing the values of three numbers that, when cubed and added together, produce the number 42 as the result.

What makes this seemingly simple problem and solution such big news is that it completes a 65-year old challenge in maths to determine whether every integer from 1 to 100 can be found as the result of such a sum of cubes. Until last week, every other number within this range has either had three integer values found that can produce them or has been determined that no such combination of integers exist to produce them.

All except for the number 42, which has now yielded to a joint effort between the University of Bristol’s Andrew Booker and MIT’s Andrew Sullivan.

The discovery is also a demonstration of the role that social media has in propagating news of an event. The same discussion thread also contains the initial inquiry that led to Numberphile’s news-breaking video describing the achievement being produced and posted the next day:

The discussion thread also contains some of the first independent checking to confirm the result, which was remarkably easy for this particular problem.

The hard part was finding the three values that would work together to produce the result, the story for which the University of Bristol told via a press release. Picking up the story following Booker’s development of an improved method for solving the problem, which had led to the discovery of a sum of three cubes for the number 33 after several weeks of effort using a supercomputer earlier in 2019:

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.

Professors Booker and Sutherland’s solution for 42 would be found by using Charity Engine; a ‘worldwide computer’ that harnesses idle, unused computing power from over 500,000 home PCs to create a crowd-sourced, super-green platform made entirely from otherwise wasted capacity.

The answer, which took over a million hours of calculating to prove, is as follows:

• X = -80538738812075974
• Y = 80435758145817515
• Z = 12602123297335631

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.

If you want to check the results for yourself, copy and paste the following expression into the Online Big Number Calculator:

(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3

The odds of Douglas Adams identifying 42 as an answer demanding a question and anticipating what kind of computer would be needed to find the question to correspond to it in Hitchhiker’s Guide to the Galaxy will likely take an Infinite Improbability Drive to resolve.

Previously on Political Calculations

• And Then There Was 42