# Sum of three cubes for 42 finally solved—using real life planetary computer

Hot on the heels of the ground-breaking 'Sum-Of-Three-Cubes' solution for the number 33, a team led by the University of Bristol and Massachusetts Institute of Technology (MIT) has solved the final piece of the famous 65-year-old maths puzzle with an answer for the most elusive number of all—42.

The original problem, set in 1954 at the University of Cambridge, looked for Solutions of the Diophantine Equation x3+y3+z3=k, with k being all the numbers from one to 100.

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.

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.

Fast forward to 2019 and Professor Andrew Booker's mathematical ingenuity plus weeks on a university supercomputer finally found an answer for 33, meaning that the last outstanding in this decades-old conundrum, the toughest nut to crack, was that firm favourite of Douglas Adams fans everywhere.

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 , 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.

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.

"So, we might find what we're looking for with a few months of searching, or it might be that the isn't found for another century."

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Bristol mathematician cracks Diophantine puzzle

More information: Andrew R. Booker, Cracking the problem with 33, Research in Number Theory (2019). DOI: 10.1007/s40993-019-0162-1

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Sep 06, 2019
Is there any pattern in the solutions? Why just 0 - 100? Why not 0 - 1000? Us human beings with our focus towards base 10 arithmetic would choose 100, while a creature with 4 fingers would probably choose 64 while a creature with 6 digits would pick 144.

Have solutions been discovered for 101, 102, or any higher numbers that 100?

Sep 06, 2019
Phew !
The universe did not collapse into a singularity.

Sep 06, 2019
@Parsec, I'm only guessing here, but I suspect they wanted solutions from 1 to 100 because those values work for percentages.

Sep 06, 2019
Have solutions been discovered for 101, 102, or any higher numbers that 100?

100: (7, -6, -3)
105: (8, -7, -4)
107: (10, -1, 2)
118: (6, -5, 3)
119: (10, -2, 3)
126: (-9, 8, 7)
132: (8, -6, -4)
135: (6, -4, -3)
144: (8, -7, -5)
153: (9, -8, -4)
154: (8, -6, -3)

101 with four cubes:
(10, 9, -8, -6)
102 with four cubes:
(10, 7, -6, -5)
But slowly, over many years, each value of k was eventually solved for (or proved unsolvable),
Hard to believe any are unsolvable, with an infinite number of cubes to try..? Seems like there'd be a combinatoric proof for the contrary, also there are an infinite number of primes ...

Sep 06, 2019
So often, a throwaway line in an article can invoke a massive subject.
This article speaks of "Charity Engine", a "worldwide computer" of PC's harnessed effectively in tandem to solve problems. I never heard of "Charity Engine", I never saw it mentioned in the news, I was never contacted to join. Was anybody asked to join or were their computers hijacked. When I leave my computer, the screen goes blank but the lights flash rapidly, as if it's being used for something!
That bring ups the subject of Bitcoin. The base on which Bitcoin is established is not money in a bank account. The most of a definition ever given was that it was an amount of computer power, apparently, from cooperating or hijacked computers, being devoted to problems. Are individuals taking control of people's computers for undisclosed purposes? Is Bitcoin based on this?

Sep 06, 2019
The Charity Engine: "super-green platform made entirely from otherwise wasted capacity."

Sorry, but this is just not true. There is no free lunch. Any modern PC consumes much less power when idle. Having your PC busy using 100% of its CPU for Charity (or BOINC) applications when you are not using it will increase your electricity consumption.

I'm a fan of BOINC and most of the distributed tasks they do, but you can't claim that it's green.

Sep 06, 2019
Actually, the screens use more power. If you want to save power on your PC turn on screen blanking.

Have solutions been discovered for 101, 102, or any higher numbers that 100?
@Proto showed some... but there would be numbers between 100 and 1000 that would take longer than the age of the universe to find solutions for, using current computing techniques. I'm not particularly sanguine about waiting 13 billion years+ for an answer.

Sep 06, 2019
Is Bitcoin based on this?

No. While Bitcoin mining is often bundled in malware, Charity Engine appears to be derived from Boinc, which is a very popular distributed computing platform. The most famous application is SETI@Home which has been searching for aliens for many many years now. https://setiathom...eley.edu

Sep 06, 2019
homestly, i lack the competency to evaluate any of the math

howeverm over the years i gave seen many variations of ponzi schemes
from timeshare offers to multi-level marketing to blatant pyramid rackets to short-selling stocks & commodities & even a couple of offers to invest in good old fashion treasure hunting

frankly & ernestly this recent enthusiasm for bitcoins has me thinking "Tulip Craze"

perhaps i'm just being a grumpy old cuss?
but damned if i can see any good ending for bitcoins

when enough people try to cash-out?
to find the vaults empty
& like "Roaring Twenties"
discovering that the bankers had boogied out of town in the middle of the night
carrying satchels of their ex-customer's deposits!

Sep 06, 2019
oops, scratch the solutions with 10, used 10^2 instead of 10^3

Sep 07, 2019
It hardly took a million hours of calculating to prove, it probably didn't take one second. It took a million hours of calculating to find.

Sep 07, 2019
It hardly took a million hours of calculating to prove, it probably didn't take one second. It took a million hours of calculating to find.
The proof refers to the entire Diophantine problem itself. Calculating 42's cubes was a necessary component of that proof, and a million+ hours were entailed by that calculation and all the preceding calculations before it.

Sep 07, 2019
Windows Calculator gets it right 42, but Excel is way off.

Sep 07, 2019
Is there any pattern in the solutions? Why just 0 - 100?

I think the article clearly answers that with "no" respectively "It's a huge start, and it was damned hard".

We do know that there are problems that will be unsolvable, and that is why probabilities based on estimates are used to focus interests.

Sep 07, 2019
- redacted -

Sep 08, 2019
Is there any pattern in the solutions? Why just 0 - 100? Why not 0 - 1000? Us human beings with our focus towards base 10 arithmetic would choose 100, while a creature with 4 fingers would probably choose 64 while a creature with 6 digits would pick 144.

The numbers 1 to 100 (or even 0 if all three variables are zero) are just a very small subset of solutions for this cubic Diophantine Equation. Because it's cubic negative numbers can also be used, thus most of the first 100 numbers can be produced as solutions. However, also because it's cubic numbers go up very fast. If you need supercomputers to find the first 100 values you can imagine what would be required to find the first 10,000 values...
Have solutions been discovered for 101, 102, or any higher numbers that 100?
You can easily try it yourself (e.g. at Google's calc) : 4^3 + 5^3 + 6^3 = 405 | -9^3 + 8^3 + 11^3 = 1114 Cubing and adding integer numbers is not difficult. The difficulty lies ...

Sep 08, 2019
(cont.) ... in alternating negative and positive numbers in every possible combination so that you find all the solutions of the range you have defined (say 1 to 1000). If you just use positive numbers your results will soon begin to explode, leaving huge gaps between them. That's why you require a supercomputer.
As you read in the article in order to find a very small number (42) they added three variables 17 numbers long each. Despite being huge numbers they are trivial to add once you've found them but no human or normal computer could have discovered them by trying all numbers up to them.

Sep 09, 2019
42 is also the number of lines on a hand-written page of the first 5 books of the Bible, as formally prescribed on every Torah Scroll used for Jewish worship.

Sep 09, 2019
So, using the first thousand cubes, some combinatorics, and some python, there are plenty of gaps but there are multiple solutions for quite a few numbers, e.g., 125 has 9 different solutions (using just the first thousand cubes):

101: (330,-272,-251)
101: (625,-621,-167)
105: (8,-7,-4)
106: (5,-3,2)
107: (51,-48,-28)
109: (45,-44,-18)
109: (124,-107,-88)
109: (901,-890,-298)
111: (892,-881,-296)
115: (11,-10,-6)
116: (5,-2,-1)
117: (962,-896,-555)
118: (5,-2,1)
118: (6,-5,3)
118: (55,-46,-41)
118: (211,-205,-92)
119: (7,-6,-2)
119: (15,-14,-8)
120: (664,-530,-524)
123: (38,-37,-16)
125: (6,-4,-3)
125: (45,-40,-30)
125: (132,-123,-76)
125: (206,-164,-163)
125: (346,-307,-232)
125: (453,-436,-216)
125: (461,-460,-86)
125: (720,-690,-355)
125: (860,-690,-675)
126: (7,-6,-1)
126: (13,-12,-7)
127: (127,-118,-74)
128: (7,-6,1)
128: (28,-24,-20)
128: (85,-77,-54)
128: (196,-188,-96)
128: (379,-371,-150)
128: (652,-644,-216)
133: (93,-91,-37)
134: (5,2,1)
. . .

Sep 09, 2019
ETA --

Forgot to have it check the (+,+,-) combination, added it, the first solution encountered (with 100 < sum < 1001) is the obvious one it missed above for 132:
132: (5,2,-1)
The tail end of the list of solutions looks like this (1000 has 12 solutions):
. . .
979 (24,-22,-13)
979: (488,-429,-334)
980: (8,7,5)
980: (11,-7,-2)
980: (31,-28,-19)
980: (224,-223,-53)
981: (10,-3,2)
981: (13,-12,8)
981: (13,-10,-6)
981: (29,-26,-18)
981: (186,-170,-115)
983: (18,-17,4)
984: (76,-74,-32)
987: (11,-7,-1)
988: (65,-61,-36)
989: (11,-7,1)
989: (14,-12,-3)
989: (197,-194,-70)
989: (476,-440,-283)
990: (11,-6,-5)
991: (10,-2,-1)
991: (12,-9,-2)
991: (13,-11,5)
991: (650,-632,-281)
993: (10,-2,1)
996: (11,-7,2)
998: (12,-9,-1)
999: (111,-106,-56)
1000: (12,-9,1)
1000: (12,-8,-6)
1000: (19,-18,-3)
1000: (90,-80,-60)
1000: (205,-200,-85)
1000: (264,-246,-152)
1000: (372,-323,-261)
1000: (412,-328,-326)
1000: (667,-648,-291)
1000: (692,-614,-464)
1000: (906,-872,-432)
1000: (922,-920,-172)

Sep 09, 2019
All these comments, and NOBODY has said anything along the lines of wondering at the coincidence that the last of this set of solutions matches that "Answer to life, the universe, and everything" that Douglas Adams chose? Just a leeetle freaky, don't you think? Thanks for the note on the lines in the Torah -- Did Adams know about that? Did he pick 42 out of a hat (at random)?

" the more interesting answers—if indeed they existed—could not possibly be calculated, so vast were the numbers required." They might have explained this more, or better. It gives the impression that super large (positive) numbers are needed, like googols or googolplexes, but I think it refers to the vast "search space" including very small numbers.

It reminds me of the Mandelbrot set, which also has many easy answers and many which quickly head off toward practical unsolvability -- has someone produced similar graphs using it?

Sep 09, 2019
Oh here we go: https://www.answe...e_Galaxy
" Douglass Adams actually sat down with a friend of his trying to come up with what they thought was the funniest number, and ended up deciding said number was 42. ... There are 42 Laws to the game of Cricket, which Douglas loved. Could it just be a coincidence? Cricket appears in several forms in the series. 101010 binary = 42 in decimal. Maybe that had something to do with it?"
But as far as Adams was consciously aware, it just sounded funny!
Of course, since then, it's been popular to throw in a reference to 42 in sci-fi and other nerdy works, but I wonder what other previous and real life appearances this number may have made.