Prize crossnumber, Issue 09

Our original prize crossnumber is featured on pages 54 and 55 of Issue 09.


  • Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
  • One randomly selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Word DVD, a dodecaplex puzzle and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide. Find out more at
  • To enter, submit the sum of the across clues via this form by 9 September 2019. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 28 September 2019.

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Dear Dirichlet, Issue 09

Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the prof’s help? Send your problems to

Dear Dirichlet,

My husband and I have found ourselves in a long-distance relationship. His company offered him a large promotion if he moved to Canada for six months, but it seems now that the position will require more time. I’m not sure that I want to move out there with him, or that I could be happy knowing he had to move back here. Conversation with the time difference is hard as well. Any tips?

— Getting tensor, Four Oaks

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Horoscope, Issue 09

Mar 21 – Apr 19

Your research output doesn’t look so bright until a tall, dark, handsome stranger presents you with a proof of the Riemann hypothesis.
Apr 20 – May 20

The heavens are not in your favour, and people may try to take advantage of you. Don’t let them take you for a mug.
May 21 – Jun 20

Tomorrow you will wake up in a parallel universe which is identical to this one, except for the fact that no one ever invented the Banach–Tarski paradox. So you get to invent it! Good for you!

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Top Ten: Chalkdust regulars

This issue, Top Ten features the top ten Chalkdust regulars! Then vote here on the top ten issues of Chalkdust for issue 10!

At 10, it’s the page you probably didn’t use to find this page: the contents page.
At 9, it’s the big block of text on the page that no-one looks at: the editorial.
Did you know that the online vote is the reason that did you know made it to number 8?
At 7, but about to move to 14 as an infinite number of people have arrived, it’s Hilbert’s hotel: the game.
At 6, it’s the puzzles page. Can you work out why it’s so popular?
At 5, and causing a moderate amount of recursion, it’s top ten.
At 4, despite being deemed not hot, it’s what’s hot and what’s not.
Storming back into the top ten after not being seen since issue 03, it’s the horoscope.

Dear Dirichlet,

what is the second most popular Chalkdust regular?

Dirichlet says: No idea.

Topping the pops this issue, it’s the crossnumber.

On the cover: Harriss spiral

The golden ratio (1.6180339…) has a rather overblown reputation as a mathematical path to aesthetic beauty. It is often claimed that this number is a magic constant hidden in everything from flowers to human faces. In truth, this is an exaggeration, but the number does however have some beautiful properties.

The golden ratio, often written $\phi$, is equal to $(1+\sqrt5)/2$, and is one of the solutions of the equation $x^2=x+1$. The other solution of the equation is $(1-\sqrt5)/2$, or $-1/\phi$. One of the nicest properties of the golden ratio is self-similarity: if a square is removed from a golden rectangle (a rectangle with side lengths in the golden ratio), then the remaining rectangle will also be golden. By repeatedly drawing these squares on the remaining rectangle, we can draw a golden spiral.

Left: The large rectangle is golden. If a square (blue) is removed, then the remaining rectangle (green) is also golden. Right: A golden spiral. Image: Chalkdust.

Numbers that are a solution of a polynomial equation with integer coefficients are called algebraic numbers: the golden ratio is algebraic as it is a solution of $x^2=x+1$. At this point, it’s natural to wonder whether you can create interesting spirals like this with other algebraic numbers. Unsurprisingly (as otherwise, we wouldn’t be writing this article), there are other numbers that lead to pretty pictures.

The plastic ratio, $\rho=1.3247179…$, is the real solution of the equation $x\hspace{.5pt}^3=x+1$. Its exact value is

A plastic rectangle—a rectangle with side lengths in the plastic ratio—can be split into a square and two plastic rectangles. If this splitting is repeated on the smaller plastic rectangles and two arcs are drawn in each square, a spiral is formed. These particular arcs are chosen so that they line up with the corresponding arcs drawn in the smaller rectangles.

Left: The large rectangle is plastic and can be split into a square (blue) and two plastic rectangles (red) and (green). Centre: The two arcs drawn in each square. Right: A Harriss spiral.

This spiral is called the Harriss spiral, and is named after its creator Edmund Harriss. It is the shape that appears on the cover of this issue of Chalkdust, and we think its resemblance to a tree in bloom makes it perfect for spring-time. We also believe that its beauty shows that the golden ratio is a gateway into a world of mathematical creativity, not an end point. There must be other nice algebraic spirals out there, buried in the roots of polynomials. If you unearth a prize-winning specimen, let us know. You may even see it on the cover of a future issue!


Page 3 model: Game of Thrones

Be warned: this article is dark and full of spoilers.

One of the best parts of getting into a series is getting to know and love the main characters. However, in Game of Thrones (or A Song of Ice and Fire, for you purists), this can be a heart-breaking activity. Who will survive to the end and who will bite the dust? No one knows, but perhaps maths can lend a hand.

Image: Andrew Beveridge

Andrew Beveridge and Jie Shan used network theory to investigate who the main characters of Game of Thrones are. The diagram above shows all the interactions between characters during the seventh series: the larger characters are more central, as determined by the PageRank algorithm.

However, it only takes one swing of an axe to drastically change the network…