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Christmas conundrum #2

It’s time for the second Chalkdust Christmas conundrum. But first of all, we can proudly announce last week’s winners. There were 82 entries to last week’s competition, of which 67 were correct. The randomly selected winners are:

  • Mike Fuller
  • Stewart Robertson
  • Catriona Shearer
  • Steven Peplow

Congratulations! Chalkdust T-shirts are on their way! The solution to last week’s puzzle can be found at the bottom of this blog post.

Now on to today’s puzzle. Four lucky people who submit the correct answer to the puzzle will win a copy of The Indisputable Existence of Santa Claus by Hannah Fry and Thomas Oléron Evans. If you want to know how great this book is, you can read our review of it here. The deadline for entries is Friday 15 December at 6pm.

Click here to download a printable PDF of this week’s puzzle

In the Christmas tree below, the rectangle, baubles, and the star at the top each contain a number. The square baubles contain square numbers; the triangle baubles contain triangle numbers; and the cube bauble contains a cube number.

The numbers in the rectangles (and the star) are equal to the sum of the numbers below them. For example, if the following numbers are filled in:

then you can deduce the following:

With the information given in the tree, you can work out the rest of the numbers.

Click here to download a printable PDF of this week’s puzzle

Once you have solved the puzzle, enter the number in the star at the top in the form below for a chance to win. The deadline for entries is Friday 15 December at 6pm. The winners will be announced in next week’s conundrum post, when you will also have a chance to win a copy of The Element in the Room by Helen Arney and Steve Mould.

The solution to conundrum #1

In last week’s conundrum, you were asked to find out who the icosahedral present was for. Stop reading now if you don’t want to know the answer yet.

Clue 5 told you that “the edges of Dominika’s present have an integer length, and her present has an integer volume”. The only Platonic solid that satisfies this is the cube, so that must be for Dominika.

Clues 1 and 2 tell you that Atheeta’s present has more faces than Emma’s, but fewer vertices. There are only two possible pairs of presents that satisfy this: the octahedron and the cube; or the icosahedron and the dodecahedron. As the cube is already taken by Dominika, the icosahedron must be Atheeta’s and the docedahedron must be Emma’s.

Finally, clues 3 and 5 tell us that the tetrahedron is Bernd’s and the octahedron is Colin’s.

So, the owner of the icosahedron was Atheeta.

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Christmas conundrum #1

It’s finally time for the first Chalkdust Christmas conundrum. Four lucky people who submit the correct answer to the puzzle will win Chalkdust T-shirts. The deadline for entries is Friday 8th December at 6pm.

It’s nearly Christmas and you have wrapped up five presents for your five best friends: Atheeta, Bernd, Colin, Dominika and Emma. You are especially happy this year as each present is the shape of a different Platonic solid.

But in your excitement, you have just forgotten which present is for which friend. You can only remember the following facts:

  1. Atheeta’s present has more faces than Emma’s present.
  2. Atheeta’s present has fewer vertices than Emma’s present.
  3. The faces of Colin’s present are triangles.
  4. Three faces meet at every vertex of Bernd’s present.
  5. The edges of Dominika’s present have an integer length, and her present has an integer volume.

Who is the icosahedral present for?

Once you have solved the puzzle, enter your answer below for a chance to win. The deadline for entries is Friday 8th December at 6pm. The winners will be announced in next week’s conundrum post, when you will also have a chance to win a copy of The Indisputable Existence of Santa Claus by Hannah Fry and Thomas Oléron Evans.

This competition is now closed.

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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.

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 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.

Continue reading

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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 deardirichlet@chalkdustmagazine.com.

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

Continue reading

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Top ten vote issue 06

What is the best mathematical celebration day?

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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.”