Prize crossnumber, Issue 06

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


  • 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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On the cover: Euclidean Egg III

Throughout my life I have made an informal study of natural phenomena, through drawing or just looking, in a spirit of curiosity. This long but unsystematic practice has given me an impression of the world around us as a dynamic and fertile system, driven by a ubiquitous tendency for spontaneous pattern formation (best understood in terms of the laws of physics) mitigated by an equally strong tendency for seemingly random variation.

It could be argued that the evolutionary process itself is driven by this tension between pattern and randomness, structure and chaos, order and disorder, theme and variation; without random mutation there would be stasis.

A bilaterally symmetric scorpion. Image: Rosa Pineda, CC BY-SA 3.0

In nature, we often see this ordering principle manifest itself as various kinds of symmetry or repetition.  Most animate creatures exhibit external bilateral symmetry; insects, crustaceans, fish, birds and animals including ourselves all tend to be bilaterally symmetric.

In common with other sentient creatures, we humans navigate and comprehend the world both spatially and temporally through pattern recognition, and being highly social creatures we are particularly attuned to reading expression and meaning in faces and bodies. It is therefore no surprise that bilaterally symmetric shapes seem to have a unique sense of potential meaning and emotional impact for us.

Whilst mirror image symmetry gives structure, the actual pattern being reflected is often far more chaotic. Like a kaleidoscope, the coloured shards are arranged at random; order is created by repetition of these random arrangements. Think of the patterns on moths, butterflies, shield bugs, ladybirds and beetles, there is often very little order in the arrangement of marks on one half, the exquisitely satisfying order of the whole is created by reflection.

Euclidean Egg III, our featured cover art this issue. Image: Peter Randall-Page

In the ‘Euclidean Egg’ series of drawings, as with much of my other work, I have chosen to use a working process which has an inherent element of chance and randomness.

There are two ordering principles in these drawings: one is bilateral symmetry, the other is Euclidean geometry. I constructed a series of geometric egg shapes in such a way as to create a seamless curve where two arcs meet. The result is a faint line drawing of an egg shape together with the construction lines needed in order to create such a taut and smooth curve. These geometric eggs by their very nature have mirror image symmetry around a vertical axis.

Folding the paper along this vertical axis and using paint introduces an element of chance. Using a pipette dropper, I spread ochre paint onto one of the areas between the construction lines on one half of the drawing. Folding the paper in half along the axis of symmetry creates two identical blobs of paint which, whilst roughly contained within the construction lines, inevitably have a somewhat random outline, reminiscent of the inlets and peninsulas of a Scandinavian island. I then add another blob of paint and continue the process, gradually building the drawing; blot by blot, fold by fold.

This process is akin to the psychoanalytic evaluation technique developed by the Swiss psychoanalyst Hermann Rorschach in 1921. Rorschach’s theory was predicted on our psychological sensitivity to bilateral symmetric shapes. He developed a series of 10 mirror image ink blots which are shown to the subject, who is then asked to say what they see in them. Their observations are then used as a way of analysing the subjects subjective response to what are effectively totally random, but highly symmetric, shapes.

Rorschach’s ink blot test has gone in and out of favour as a psychoanalytic tool during the last century but for me, our reaction to his ambiguous symmetric forms reveals something about the way in which our perception of the world is driven by subjective projection of feeling as well as objective analysis and observation. We read meaning into the world as well as taking meaning from what we perceive.

A construction of the simplest Euclidean egg

My fundamental concern in making art is an exploration of what makes us tick, the emotional subtext to our everyday experience.  The world enters our consciousness as emotion and expression as well as information and knowledge. We respond to shapes and colours, forms and spaces, poetry and music in ways which can be difficult to analyse or quantify.
Whilst we have so many ways of communicating with one another (not least language itself), the medium of visual art is uniquely capable of exploring these often intangible emotional responses.

In this particular drawing I am attempting to reconcile order and randomness, Euclid and Rorschach. My attention is concentrated on making a satisfactory balance between the ‘theory’ of pure abstract geometry with the ‘practice’ of what happens in the real world (in this case, the viscosity of the paint as well as the texture and absorbency of the paper are all determining factors).

Being preoccupied with my attempt to reconcile these polarities is strangely liberating. The task involves innumerable decisions and appraisals which is conducive to a spontaneous and playful approach. In fact, play is an important concept for me. Play can be unselfconscious and create fresh associations and ideas. In order to play well, however, one needs a playground. Football without rules and a finite pitch would neither be fun to play nor interesting to watch.

Although rooted in a study of natural phenomena, my work is less concerned with reproducing existing forms than with trying to grasp the underlying dynamics which determine the shapes and forms we see around us and to use these dynamic processes to create new objects which are both novel and familiar.

In the words of the philosopher and art historian Ananda K Coomaraswamy in his 1956 essay The Transformation of Nature in Art, “art is ideal in the mathematical sense like nature, not in appearance but in operation.”


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.