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Prize crossnumber, Issue 05

Our original prize crossnumber is featured on pages 58 and 59 of Issue 05.

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. 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, with free UK shipping. Find out more at mathsgear.co.uk
  • To enter, submit the sum of the across clues via this form by 22 July 2017. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 30 July 2017.

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On the cover: dragon curves

Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper $n$ times, then the edge will make an order $n$ dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of issue 05 of Chalkdust, and in the header box above, is an order 10 dragon curve.


Left: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Right: A level 10 dragon curve resembling a dragon


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Dear Dirichlet, Issue 05

Dear Dirichlet

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

Dear Dirichlet,

I personally have a very deep, long-held belief in free-market capitalism and the value of hard work, but I was recently shocked to discover when I switched subjects that most people in my new research area are staunch followers of Karl Marx! I’ve had many arguments with my new colleagues on this. I can feel my energy slowly draining with every passing debate. How do I resolve this?

— Feeling blue, Surreyv

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Page 3 model

Bread is a staple of many diets. From delicious garlic bread to crunchy pizza, it’s enjoyed throughout the world. But have you ever wondered what mathematics lies just beneath the crust?  Thankfully DR Jefferson, AA Lacey and PA Sadd at Heriot-Watt University have! No? Well, we’re going to tell you anyway.

                   

Bread dough is initially a bubbly liquid, with bubbles connected to other bubbles in a ‘matrix’.  These bubbles will collapse, provided that both the temperature and temperature gradient are high enough. To start with, the bubbles at the surface (which is hotter than the interior) reach a temperature at which they are likely to fracture. At this point, the temperature gradient is also high, with plenty of cooler liquid dough nearby. However, when the temperature of the interior has increased sufficiently to allow the bubbles inside to burst, the temperature gradient is much lower, the matrix has set, there is less liquid dough nearby, and so less collapse can take place.

                  

But that’s not all! We can refine the model by considering the movement of the ‘crust boundary’ (where bubbles collapse) as the dough rises, as well as the vaporisation of moisture inside the bubbles. Both of these allow for the transfer of heat and affect the thermodynamics of the whole process.

                    

So in the future, please try to remember all the maths that worked hard to ensure the crustiness of your bread! And, on that note, we’re off to get pizza…

References

Jefferson DR, Lacey AA & Sadd PA 2007 Crust density in bread baking: Mathematical modelling and numerical solutions. Applied Mathematical Modelling 31 (2) 209–225.
Jefferson DR, Lacey AA & Sadd PA 2007 Understanding crust formation during baking. Journal of Food Engineering 75 (4) 515–521.