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Christmas puzzle #1: Christmas tree sudoku

Here at Chalkdust, we like to celebrate Christmas as much as the next magazine for the mathematically curious, and what better way to celebrate than with a few yuletide mathematical puzzles. We have three for you, the first one you can find below, the second one will be published tomorrow (Christmas Day), and the final one the day after (Boxing Day). They are the perfect accompaniment to an warming hot chocolate and mince pie. Each puzzle is related to the previous one, so keep a hold of your solutions ready for the next day. We hope you enjoy giving them a go and the whole team wishes you a very merry Christmas!

The rules

  • Normal sudoku rules apply: you must complete the 9×9 grid with the digits 1 to 9 such that each digit appears exactly once in each row, column, and 3×3 block.
  • The digits that appear on each thermometer must strictly increase as you move away from the bulb. The colours of the thermometers are purely decorative and do not affect the puzzle.
  • The digits on the baubles are all even.
  • The digits on the stars are all prime.

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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 14 March 2021. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 May 2021.

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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.