33 and the sum of three cubes

You may have seen that 33 can be expressed as the sum of three cubes.

They are (8866128975287528)^3 + (-8778405442862239)^3 + (-2736111468807040)^3 = 33

The solution was discovered by Andrew Booker from the University of Bristol.

Check out his paper.

If I may quote Andrew’s abstract (ahem!)…

Inspired by the Numberphile video “The uncracked problem with 33” by Tim Browning and Brady Haran, we investigate solutions to x^3+y^3+z^3=k for a few small values of k. We find the first known solution for k=33.

In all the excitement, some people mistakenly attributed the finding to Tim Browning (formerly of Bristol but now in Austria), who was helping publicise Andrew’s result. Tim had introduced many people to the problem via the Numberphile video.

A embarrassed Tim has been quick to remedy this, pointing out Dr Booker’s paper.

33 was previously the lowest number for which the sum of three cubes status was unknown.

There are lower numbers which can never be expressed as the sum of three cubes (4, 5, 13, 14, 22, 23, 31, 32 ) but these were known to be impossible because they can be represented as 9k+4 or 9k+5.

Until now, 33 was the lowest number which was unknown.

There is yet to be a formal proof that all numbers (outside the 9k+4 9k+5 category) can be expressed in this way.

For more on this, see the Numberphile video: The Uncracked Problem with 33 - Numberphile

PS: This is the second breakthrough inspired by the 33 video.

How many points do you need in the Premier League?

We’ve run a computer simulation of one million football (soccer) seasons.

The aim was to find out how many points you need to secure the most important positions in England’s Premier League.

  • How many points will win it?

  • How many points secure a coveted top four position (and entry to the Champions League)?

  • How many points avoid the bottom three, and dreaded relegation from the league?

The simulation was done by Professor Tony Padilla and Dr Adam Moss from the University of Nottingham’s Physics Department.

We did it for this Numberphile video, which outlines the results:

It’s technically possible to win the league with 38 points (all 380 games in a season are drawn, goal difference is tied but your team scores the most goals).

Likewise, it’s technically possible to be relegated despite scoring an impressive 63 points (the top 18 teams win 21 and lose 21 games each, but your goal difference is the worst).

But we thought a Monte Carlo simulation was a better way to explore more realistic possibilities.

For the million seasons we re-created, exact scores were generated for each individual game. That’s 380 million results.

Data from last season was used to weight the teams’ strengths rather than treating them all equally (more info on that here).

It is perhaps better to consider our simulation as a million “re-runs” of a season rather than predicting the future. Much can change in one million years!

But it still provides fascinating insights into how many points are required.

Here’s the number of points scored by the teams in each league position, across the million seasons:

The following is evident from the totals above:

A few other bits of trivia:

  • A perfect season (ie: 38 wins) was achieved 62 times out of the million simulations.

  • The league was also tied on 62 occasions - with points, goal difference and goals scored unable to decide the winner!

  • In one season the league winner scored just 50 goals (must have been a lot of 1-0 wins).

  • The most goals scored in a single season was 161.

  • There was one freakish instance of team scoring just 22 points in a season and NOT being relegated.

So which teams were winning and losing across the million seasons?

(You should be able to enlarge the image below by clicking)

  • No surprises that Manchester City won the league nearly 90% of the time.

  • Liverpool were next best, despite our work being based on last season in which Liverpool finished 4th. This is most likely due to Liverpool’s goal-scoring power.

  • No team outside the “Big Six” ever won the league, and Chelsea was the only “Big Six” team to be relegated (on just five occasions out of a million).

  • Southampton, despite finishing a lowly 17th last year, performed surprisingly well, even finishing second in one season.

Finally, a few single-game novelties from the 380 million simulated games. These seem unlikely, but remember they’re over the course of ONE MILLION SEASONS.

  • Most goals in a single game: Manchester City 17 Everton 5

  • Highest scoring draw: Everton 9 Liverpool 9

  • Biggest winning margin: Manchester City 20 Bournemouth 1

  • Highest score by a losing team: Everton 11 Chelsea 10

Thanks to Jeremy Nicholas for being stadium announcer in our main video.

The Numberphile Podcast

I speak with many fascinating people while making Numberphile videos.

So it seems to make sense to have a podcast where I can talk to them in more depth.

The first episode is an interview with fellow math YouTuber, Grant from 3blue1brown.

Plenty more coming, including fun episodes Hannah Fry and Cliff Stoll --- plus all sorts of world-famous mathematicians.

Podcast webpage: https://www.numberphile.com/podcast/
iTunes: https://itunes.apple.com/gb/podcast/the-numberphile-podcast/id1441474794
Overcast: https://overcast.fm/itunes1441474794/the-numberphile-podcast
RSS: https://www.numberphile.com/podcast?format=rss

Plus its gradually finding its way onto other platforms and players - so check yours.

Please do subscribe on your podcast player.

Another Periodic Videos Top 10

For Periodic Videos’s 10th anniversary, we published “top ten” lists by both myself and Professor Sir Martyn Poliakoff.

Here’s another top 10 supplied by one of our viewers and a good friend of the show, Andres Tretiakov. He’s a technician at an English school and a real chemistry enthusiast.

Andres and The Prof

Andres and The Prof

He avoided any duplication with videos on mine or Sir Martyn’s list…

Tsar's Vodka and Gold


I love Aqua Regia and especially all the history and alchemy associated with it. The red colloidal solution or sol of gold nanoparticles at the end also demonstrates that red stained glass in churches contained gold nanoparticles.



Neil compilation: because he is the MASTER and I can only aspire to be like him.

Carbon Dioxide Cannon


 Because by making mistakes science moves forward!!!

Periodic Table Clock (12 Days of Christmas)


The whole series of the 12 Days of Christmas is a pleasure to watch. It really shows just how far away Periodic Videos has reached and the wonderful response of the fans that love chemistry across the globe. In addition, it made me reach out to Nagayasu Nawa and congratulate him for such beautiful artwork, clocks, fans and PT Happi. Since then, we have exchanged gifts, ideas and suggestions for a few years now and I consider him a good friend thanks to Periodic Videos.

Fire Water


Any reaction involving potassium metal is a favourite of mine. Here Sam Tang showed me a really cool demo that I have used many many times and for that I’m very grateful! It’s beautiful and there is so much chemistry going on!



 Not only this one but ALL the videos involving trips around the world where the Prof held meetings, sharing and discussing chemistry in a more personal approach.

Dynamite and TNT


Being an enthusiast of high energetic materials I thought this video was great! The supersonic wave shown in the photos and the history described is striking. I immediately bought the book Canary Girls of Chilwell.

Pouring Mercury into Liquid Nitrogen (slow motion)


Because sometimes you have to try things out!!!! The scientific method in action!

How much caffeine in coffee?


Because we are really addicted to caffeine and it’s my favourite molecule! I have extracted caffeine from coffee, yerba mate (from Argentina), tea and chocolate powder. A very good introductory organic chemistry experiment.

Nanoparticle Sign


I think this was my first gift to the Prof and to Periodic Videos back in 2011. I have fond memories of making it and a whole video dedicated to it was an unexpected pleasant surprise!

Peru Pictures

Recently back from a trip to Peru.

The trip started in Cusco, then followed the Salkantay trekking route to Machu Picchu.

I’ll have more information soon, including a podcast recorded during the trip.

But for now, here are some photos.

Belphegor's Prime and Harvey Dubner

For Halloween this year we released two videos about Belphegor’s Prime (on Numberphile).

If you’ve only seen part one, then part two is also well worth a look.

You may notice (in part one) that Tony Padilla makes mention of a “prime hunter” called Harvey Dubner.

Prior to publishing, I sent an email to Harvey because I wanted to include a picture of him in the video.

I received a reply… but unfortunately it arrived just after publication. The message came from Harvey’s son, Bob.

In a lovely email, Bob explained his father is now 90 years old.

He explained Harvey is in “pretty good” health, although suffering some memory problems and no longer does mathematics.

Bob, who was also instrumental in the prime number work, explained:

“I am, indeed, the designer of the “massive computer” we “built in the garage” – it actually was a series of software packages and special arithmetic hardware that I designed over the years between about 1981 and 2000.  They mostly sat next to the couch in the family room; Dad liked to have the TV on when he did mathematical research.

It was around the year 2000, when FFT routines  running on 400MHz and faster PCs overtook the hardware I had designed, that we gave up trying to improve my hardware.  By the year 2005, I had incorporated the GMP software package and the magically tweaked routines written by George Woltman (of the GIMPS project) into our software.  (I’m a pretty good programmer, but some of the people on the GMP project and Mr. Woltman are magicians.  And my long-time engineering motto has been, “We steal only the finest.”) 

We had a great run with the stuff I built; I believe that at our personal peak we had found about three-quarters of all the known prime numbers with more than 1,000 digits.  But by the early 2000s, not only were general-purpose computers faster than my hardware (which ten years earlier had given us supercomputer capability) but also lots of people started using publicly available software to search for big primes, purely for bragging rights.  So my father switched to looking for numbers he found interesting.  Many packages are optimized around calculating the primality of number of the form k * 2^N.  When k is one, you have the Mersenne primes, of course; but the calculations can run fast even for other values of k.  But my hardware and software was general purpose, so he looked for numbers that were interesting in base 10 – for example, the palindromic primes, of which the Belphegor Primes are a subset.  That was just for fun; he did a bunch of serious work in Sophie Germain Primes and Carmichael numbers.  But it was always fun searching for big primes of various kinds.”

Bob included a picture of Harvey, adding:

“He remembers that he used to do a lot of math, but he can’t remember the math at all.

He likes being reminded about it, though, and so I happily send along this picture, which I shot about a year ago.

Coincidentally, a friend just today pointed me at https://youtu.be/zk_Q9y_LNzg, which is, I believe, the episode in question, so I am probably too late with the picture.  But I figured I’d send it along anyway.

Thanks so much for your interest.  If you decide to edit in my dad’s picture, please let me know.  It would make him happy to see his image mentioned along with his work that way.”

So here’s the picture of Harvey - our thanks to him and to Bob.