42 Finally Solved!

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.

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Then, earlier this year, a solution for 33 was finally found — leaving as the only remaining unsolved number, the notorious 42.

Much to the delight of Douglas Adams fans everywhere. (See my post, The Number 42, if you don’t know why that number is special to us.)

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)!

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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:

-522,413,599,036,979,150,280,966,144,853,653,247,149,764,362,110,424

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.

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So crank up your computers and dig in!

Stay cubic, my friends!

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

8 responses to “42 Finally Solved!

  • SelfAwarePatterns

    “It took over a millions hours of calculating”

    That’s orders of magnitude better than Deep Thought, which took 7.5 million years to get the same answer. So much for pan-dimensional, hyper-intelligent beings who look like mice.

    • Wyrd Smythe

      Ha, yeah! That’s what they get for using MS-DOS 3.3. 😀

      (Although, in their defense, I suppose it’s a little bit harder coming up with the answer to “Life, the universe, and everything,” than finding three cubes that add up to some number.)

      • SelfAwarePatterns

        Pfft! 6 X 7. And they didn’t even know that’s what they were doing.

        On the three cubes, it is impressive that the numbers had to get that big just to land on 42.

      • Wyrd Smythe

        But the question can’t be as simple as “What do you get if you get if you multiply six by seven?” Surely it wouldn’t take the Earth computer ten-million years to come up with that!

        And what about Arthur seeing the ape-descended life form (who would eventually think digital watches are a good idea) drawing “scrabble” tiles to spell out: “What do you get if you multiply six by nine?”

        Something’s fishy about the whole thing!!

        As to the three cubes, yeah, that’s pretty wild. I wonder if crowdsourcing will cause the remaining ten three-digit numbers to fall. Or maybe they require much larger numbers, and it’ll be a while.

        I don’t know if you watched any of the videos, but apparently another version of the game is finding other answers to the solved numbers. There’s a theory that (as with primes) there are an infinite number (but they’d involve bigger and bigger cubes).

      • SelfAwarePatterns

        “Something’s fishy about the whole thing!!”

        Heresy! The sacred Hitchhiker books cannot contain a contradiction. If you think you found one, it only means a long treatise needs to be written to explain how it is, in fact, not a contradiction at all. You’ll see, once I get around to, you know, actually writing it.

        Have to admit I haven’t watched the videos. Sorry. My math geekness is just too inadequate.

      • Wyrd Smythe

        😀 IIRC, the last book (or maybe the fourth) mentions something about The Answer and The Question being mutually exclusive (like the position and momentum of a particle), so now that The Answer is known, The Question never can be. (Which may be actually why the Earth was destroyed by Vogons… reality works in mysterious ways. Probably the new Earth will also experience catastrophic failure before the computation completes.)

        BTW: The BBC videos of HHGttG are on Amazon Prime, and I’m slowing re-watching them for fun. (I believe the videos are based on the radio program, and the books are based on them, so much of the dialog is word-for-word. So many great lines!)

      • SelfAwarePatterns

        I rewatched the BBC ones several ago, not long after that movie came out. Despite lower production values, the old TV show was much better.

        I didn’t realize until a few years ago that the earliest Hitchhiker’s content was the radio show, that the books were based on them rather than the other way around. I wonder if that radio show is available anywhere.

      • Wyrd Smythe

        I didn’t hate the movie like some fans did (Bill Nighy as Slartibartfast, one of my favorite characters, was worth the price of admission), but I agree the TV series is way better.

        The radio version is available — Amazon has it. (Pricy: in the $80 range!)

        I read once that HHGttG is unique in having eleven different representations. I usually can’t remember them all, but’s something like, it’s a:

        1. Radio program
        2. TV series
        3. Book series
        4. Movie
        5. Stage play
        6. Video game
        7. Record album
        8. Audio book
        9. Comic book
        10. ??
        11. and a towel

        I think the missing one is either the live radio version (versus pre-taped), a musical stage play (versus non-musical), or a board game (versus video game). Looking briefly at Wiki, it may be more text adventure versus video game on that last one.

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