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Dear Dirichlet, Issue 12

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

Dear Dirichlet,

I have recently entered retirement, having handed over my day job—writing bafflingly popular hyper-violent thrillers which end with the villain getting crushed by an oak bookcase—to my younger brother. But I now find myself with time (as well as coffee and cigarette stains) on my hands. I’m thinking of dipping into movie making. Any good plot ideas?

— Leigh Children, now Wyoming apparently?!

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Prize crossnumber, Issue 12

Our original prize crossnumber is featured on pages 56 and 57 of Issue 12.

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. 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 Nerd DVD, and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping.
  • To enter, enter the sum of the across clues below by 3 September 2020. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 October 2020.

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Colouring for mindfulness

Imagine three mice equally distanced from each other, ie at the vertices of an equilateral triangle. If at the same time, all three mice start chasing their neighbour clockwise, then each of their paths would be a logarithmic curve. But this is rather hard to draw, especially if we want to restrict ourselves to only using a ruler.

Instead, let us imagine that the mice can only run in a straight line and need to stop to reassess their direction. If at a given stage we draw their intended path, and assume that the mice cover a tenth of the distance to the next mouse before stopping and reassessing their direction, we get the picture below. While these pictures have been drawn using straight lines only, we see three logarithmic spirals emerging:

Stages 1, 2, 3, and 20.

But why stop there? Why not start with $4$, $5$ or $n$ mice on the vertices of a regular square, pentagon or $n$-gon? The following instructions show a very algorithmic approach to drawing these patterns:

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Page 3 model: Solitons

Solitons are special analytic solutions to the nonlinear wave equations that turn up everywhere: from fluid dynamics to quantum mechanics to molecular biology. One such equation is the Korteweg–de Vries (KdV) equation for a 1D wave $u(x,t)$ evolving in time $t$: \[ \frac{\partial u}{\partial t}+6u\frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0. \] What makes solitons special is that they behave in many ways like solutions to the linear wave equation: propagating without losing their shape, and even interacting nicely:

In 1990, Daisuke Takahashi and Junkichi Satsuma proposed a discrete model for solitons called the ball and box model. It consists of an infinite row of boxes, some of which contain a ball. A sequence of consecutive balls represents a wave. The pattern of waves after each time step is found by starting from the left and moving each ball to the next available empty box to its right.

A row of $n$ consecutive balls behaves like a single wave, and moves to the right at a constant speed $n$. Several such waves, so long as they start sufficiently separated, will interact (in a possibly messy way), and then disperse, maintaining their original shapes overall.

This model is fun to play with, but—surprisingly given its simplicity—it manages to capture a large amount of the interesting and unusual behaviour of the fluid waves modelled by the KdV equation.

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Top Ten: Maths-themed days out

This issue, Top Ten features the top ten maths-themed days out! Then vote here on the top ten calculator buttons for issue 13!

At 10, it’s the scorpion enclosure at London Zoo.
At 9, why not take a punt on Cambridge’s mathematical bridge?
At 8, it’s Escher in the Palace in The Hague.
Don’t throw away your shot to visit the bridge where it happened: you’ll be back to see number 7, Broom Bridge in Dublin.
Enjoy a cracking day out at number 6, Bletchley Park: it’s the bombe!
At 5, it’s a visit to both Bletchley Park and the National Museum of Computing on the same day.
The only safe cycling experience in New York is our number 4, at MoMath: the National Museum of Mathematics.
At 3, it’s a trip to the Lake District with a calculator.

At 2, it’s one of the perfect places to walk one mile south, one mile east, then one mile north: the north pole.
Topping the charts for a record seven weeks, it’s a walking tour of Königsberg.