# Hunting for polydivisibles

Suppose we have a number in which the first digit is divisible by one (and isn’t zero), the first two are divisible by two, the first three are divisible by three, and so on all the way up to a ten digit number that is divisible by, unsurprisingly, ten. Such a number is called a polydivisible number, and I’m quite excited to be writing about them because in my opinion there isn’t enough written about these wonderful numbers.

While I’d like to focus this article on the discoveries I made while searching for the largest polydivisibles I could find, this being a maths blog I think it’s only fair I give you the opportunity to hunt for them yourselves. The challenge is to find the ten digit number I just described, using each of the digits zero through nine just once. (Spoiler alert: the solution is in a couple of paragraphs’ time.)

I first read about polydivisible numbers in Matt Parker’s brilliant Things to Make and Do in the Fourth Dimension. While he did investigate the details in a fair amount of depth, he was only able to give solutions for up to base 16—poor effort, Matt! A base is the mathematical way of expressing the number of digits available to us. Binary is made up of ones and zeroes, and so we call this base-2. When counting, we have ten digits (including zero), and therefore it is base-10 that is used in our day to day lives.

In base-10, the solution you hopefully found (don’t tell me you didn’t work it out!) is 3816547290. I’m a software developer myself and so the most interesting part for me was finding shortcuts to the answer. While a computer can find the solution by brute force fairly quickly, it still takes a bit of time as there are 3,628,800 possible answers to check: in computer science, we call these permutations.

I read a little further, and found out that indeed there were shortcuts that could be made—this is the power of mathematical thinking in problem-solving: a monumental task can become considerably easier by simply applying a little bit of thought. For example, we know that the first two digits represent a multiple of two, and therefore we only need to check permutations where the second digit is zero, two, four, six or eight. Applying this and a few other rules, I was able to solve the problem for base-10 in a mere 16 milliseconds. Perhaps saving a few milliseconds doesn’t sound too exciting, but in computing it can really make a difference. Saving a little bit of time in just one calculation, if it has to be repeated a billion times, is very beneficial. The graph below shows how much longer it takes to solve this puzzle when we add just one extra digit.

https://plot.ly/~rafaelprietocuriel/63/computation-time-spent-searching-permutations/

To finish, I did promise that I’d outdo the sixteen bases Matt was able to find—I don’t really think his effort was poor, but having a bit more development experience under my belt I was able to make optimizations leading to the discovery of all of the solutions up to base 30. After 9C3A5476B812D in base fourteen, I was able to find… no more! Perhaps base-14 is as high as you can go? So far, despite extensive research and calculations, I haven’t been able to find any number in a higher base. There’s certainly an opportunity for a proof or counterexample here.

And I’ll leave it at that—this was just one journey into a fairly abstract and arguably pointless use of numbers. The truth is, however, that it still provides a great adventure for anyone willing to embark on it.

If you’re interested in the software I used to solve this puzzle, or perhaps you’re looking to beat the base-30 I achieved, the code is available for free here.

# New largest prime number discovered!

Here at Chalkdust we’re very excited by the latest discovery of the new largest prime number, which is the Mersenne prime $2^{74,207,281}-1$. So to celebrate this discovery by the Great Internet Mersenne Prime Search, we thought we’d publish the number.

### Fun facts first:

• All Mersenne primes are of the form $2^p – 1$, where $p$ is prime (the first four are 3, 7, 31, and 127).
• Mersenne primes are named after Marin Mersenne (whose face is in the banner at the top!).
• In binary, the number is ‘1’ repeated 74,207,280 times!
• This means it requires 8.85MB of disk space to store, or 7 floppy disks!
• Using the “million, billion, trillion” naming system, you could call this number 300 septillisensquadragintaquadringentiilliquattuorducentillion!

# Pandigital square numbers

A square number containing every digit from 0 to 9 exactly once.
Chalkdust Prize Crossnumber #1, 5 down

Whilst trying to answer 5 down in the Chalkdust crossnumber, I discovered that there are 87 square numbers in base 10 that use every digit exactly once (without a leading zero). This made me wonder how many square numbers have this property in other number bases.

I quickly wrote some code to get an idea of how many pandigital (without repeating digits or a leading zero) squares there are in other number bases.

# What’s your favourite number?

Between Saturday 27th June and Sunday 5th July, Newcastle University, in association with the Institute of Mathematics and its Applications (IMA) and The Great North Maths Hub will be holding a Numbers Festival, during which they will be asking people what their favourite number is and why. Favourite numbers for many are linked to birthdays or anniversaries, or the number worn on the shirt of a favourite footballer. Below is our modest contribution to the Numbers Festival, perhaps with a slightly different take on their question… Continue reading

# What’s hot and what’s not, Issue 01

“A blackboard, for a mathematician, is an unrivalled way to communicate” – Cédric Villani, Fields Medallist 2010

Blackboards

The mathematical equivalent of a Little Black Dress. Blackboards will never go out of style.

E-learning systems

In the immortal words of the Black Eyed Peas, interactive whiteboards are so two thousand and late.

# A Fields Medal at UCL: Klaus Roth

Be proud if you are studying Mathematics at UCL! Looking back, we have numerous famous alumni who later gained significant achievements in their field.  One of them is Klaus Roth, who was once a research student at UCL, and later was a lecturer and professor at the university, during which time he won the Fields Medal.

If you haven’t heard of the Fields Medal, it is seen as the equivalent of the ‘Nobel Prize’ in Mathematics (although unfortunately it has a much lower monetary reward) and is awarded every four years by the International Mathematical Union.  The award is given to a maximum of four mathematicians each time, all of whom must be under the age of 40 and have made a great contribution to the development of Mathematics.  Roth won the Medal in 1958, when he was 33 years old and still a lecturer at UCL (show more respect to your lecturers … you never know!), for having “solved in 1955 the famous Thue-Siegel problem concerning the approximation to algebraic numbers by rational numbers and proved in 1952 that a sequence with no three numbers in arithmetic progression has zero density (a conjecture of Erdös and Turàn of 1935).”

# £100 Prize Crossnumber, Issue 01

Our original £100 prize crossnumber is featured on pages 34 and 35 of Issue 01.

### Rules

• Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0.
• One randomly selected correct answer will win £100. Three randomly selected runners up will win a Chalkdust pen. The prizes have been provided by G-Research, researchers
of financial markets and investment ideas. Find out more at gresearch.co.uk.
• To enter, email crossnumber@chalkdustmagazine.com with the sum of the across clues by 22 July 2015. Only one entry per person will be accepted. Winners will be notified by email by 1 August 2015.
• Entry is now closed.