Musicians practice; actors rehearse; athletes work out; and mathematicians play with numbers. Some of the games they play may seem as silly or pointless as musicians playing scales, but there is a point to it all. That old saying defining insanity as doing the same thing over and over and expecting different results was never really correct (or intended to be used as it often is).
An old joke is more on point: “How do you get to Carnegie Hall?” (Asked the first-time visitors to New York.) — “Practice, practice, practice!” (Replied the street musician they asked.) The point of mathematical play can be sheer exercise for the mind, sometimes can uncover unexpected insights, and once in a while can be sheer fun.
As when finally solving a 65-year-old puzzle involving the number 42!
The puzzle in question involves the Diophantine Equation:
K = X3 + Y3 + Z3
The game is to find solutions (or prove no solution is possible) for K, for the numbers from 1 to 100 (and, for extra credit, also up to 1000).
Some solutions are trivial:
3 = 13 + 13 + 13
The puzzle was first suggested at the University of Cambridge in 1954, and in the 65 years since, especially in the era of powerful modern computers, solutions have been found for most of the numbers (22 were known to have no solution, and 69 were found in 1954; only one more was found by 1963).
Until this year, two particularly tricky ones remained: 33 and 42.
Then, earlier this year, a solution for 33 was finally found — leaving as the only remaining unsolved number, the notorious 42.
Here is a 2015 video (from YouTube channel Numberphile) that describes the puzzle very well but which predates the solving of 33 (and 74 and 42).
In 2016, Andrew Booker from the University of Bristol found a solution for 74 (here’s the Numberphile video).
And here’s a 2019 video describing the solution for 33 (and which provides more details about the problem and approaches to solving it):
There is also an early write up of the Booker’s work available here.
The solution is:
X = +8,866,128,975,287,5283
Y = -8,778,405,442,862,2393
Z = -2,736,111,468,807,0403
Which all left only the number 42 (of numbers up to 100)!
And now, also this year, that magical mystical number has a solution:
X = -80,538,738,812,075,9743
Y = +80,435,758,145,817,5153
Z = +12,602,123,297,335,6313
If you look closely, you’ll see that these three numbers all have one more digit than the numbers that solve 33. This is why they weren’t found in the search that found 33.
For what it’s worth, I’ve verified both solutions are correct (as if it was even possible they weren’t). The cubes of those numbers are extremely big. Here’s the value of just X-cubed:
The other two are equally big!
Here’s the video:
One thing that’s kind of neat about this solution is that, while Booker found the solution for 33 using the University of Bristol’s supercomputer, he used crowdsourcing to find the solution for 42.
It took over a millions hours of calculating to finally solve a problem that is just about as old as I am (which is to say: quite old-ish)!
All that remains are a handful of numbers between 100 and 1000 (two handfuls, actually): 114, 165, 390, 579, 627, 633, 732, 906, 921, 975.
So crank up your computers and dig in!
Stay cubic, my friends!