Prize crossnumber, Issue 06

Our original prize crossnumber is featured on pages 52 and 53 of Issue 06.

Correction: The pdf was incorrect and 5D did not match the clues below. This has now been fixed.
Clarification: Added brackets to 29A and 34D to reduce ambiguity.


  • 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
  • To enter, submit the sum of the across clues via this form by 8 January 2018. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 22 January 2018.

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Dear Dirichlet, Issue 06

Dear Dirichlet

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

Dear Dirichlet,

I’ve just started my PhD at a well-known university, and I’m trying to make some friends. There are supposed to be 55 other students but nearly everyone in the PhD office refuses my offers of tea, sits in silence, and will barely talk to me unless I whisper them some very specific technical questions. I was hoping there would be some people in the group who enjoy everyday things: biscuits, beer, and just shooting the breeze. Is this really what academia is like?

— Pearl among swine, Withheld

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Top ten vote issue 06

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Page 3 model: Crowd control

Being part of a crowd is something that we all have to experience from time to time. Whether it’s in a busy shop or commuting to work, the feeling of being swept along by those around us is all too familiar. The ubiquity of the situation, and the huge amount of data available from CCTV footage, makes crowd dynamics a favourite subject for mathematical modelling.

One popular method is known as the social force model, which applies Newton’s second law to each member of the crowd. Each individual accelerates to maintain their ‘desired velocity’, and this is balanced against forces from physical obstacles as well as the social force that maintains polite distance between people—a mathematical interpretation of personal space!

Lanes naturally form when people walk in opposite directions. Image: Dirk Helbing and Peter Molnar

Huge simulations of up to a million pedestrians have been run, which show the model’s remarkable powers. If groups of people want to travel in opposite directions along a bridge, for example, lanes of alternating direction naturally form to minimise “bumping”.

When two crowds meet at a gap, the walking direction oscillates. Image: Dirk Helbing and Peter Molnar

Some of the results are more unexpected. For example, if people try and move too fast then it can actually slow them down via an increase in ‘friction’ that results from pushing. Further, it can be shown that two narrow doors are a more effective way of leaving a room than one big door, so putting a bollard in the middle of an exit actually speeds people up!

Still, not much solace when you’re stuck in a Christmas scramble at Woolworths…


Helbing D and Molnar P (1997). Self-organization phenomena in pedestrian crowds. In: Schweitzer F (ed.) From individual to collective dynamics, 569–577.


How to make: a hyperbolic plane

You will need

  • triangle paper
  • scissors
  • sticky tape


1. Cut out a hexagon and a triangle from the triangle paper.

2. Cut along one of the lines from a corner of the hexagon to the centre.

3. Tape the triangle between the two edges of the cut you just made. There is now more than 360° around the point, so the surface will not be flat.

4. Continue to tape more triangles to the surface, making sure there are always seven triangles at each point.

5. Congratulations! You have made a hyperbolic surface.