How to make: a Möbius Surprise

You will need

  • A3 paper
  • scissors
  • glue


1. Cut your A3 paper into 4 strips.

2. Take two strips and glue them perpendicularly to make a cross.

3. Make a Möbius strip with each of the perpendicular strips.

4. Cut along the centre line of one Möbius strip until you get to where you started. Then cut along the other.

5. Be surprised.


Prize crossnumber, Issue 08

Our original prize crossnumber is featured on pages 54 and 55 of Issue 08.


  • 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 Word DVD, a dodecaplex puzzle and much, much more. 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. Find out more at
  • To enter, submit the sum of the across clues via this form by 2 February 2019. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 16 February 2019.

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Dear Dirichlet, Issue 08

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

Dear Dirichlet,

The annual village fete is fast approaching, and every year I embarrass myself at `guess the number of sweets in the jar’. My exasperated wife ends up telling me to just say a number, and I always panic. Last year my guess was $\mathrm{i} – \text{π}$. Maybe I was just hungry.

— Hungry hungry hippo, Gospel Oak

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Top Ten: Units of measurement

This issue, Top Ten features the top ten units of measurement! Then vote here on the top ten Chalkdust regulars for issue 09!

At 10, it’s the Zappa.
At 9, and selling one tenth of the number of copies that number 6 sold: it’s a millimetre.
At 8, and not receiving much radio play due to being far longer than the rest of the top ten: it’s a furlong.
Following warm reviews from critics, degrees Celsius enters the top ten at 7.
The new single by no-one’s favourite rapper 50 Centimetre is at 6.
At 5, it’s the Yardbirds tribute act whose members are all 8.5cm taller than the originals: the Metrebirds.
Following the release of its 51st anniversary deluxe edition, The Velvet Underground and Picometre is at 4.
At 3, and selling 273.15 more copies than this issue’s number 7: it’s Kelvin Harris.
Image: Wikimedia commons user Martinvl, CC BY-SA 3.0.
Forced up two places from last issue, it’s the Newton.
Topping the pops this issue, it’s dimensionless constants.

Top ten vote issue 08

What is the best Chalkdust regular?

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Page 3 model: Ponytails

The ponytail hairstyle is a synonym of comfort and simplicity, and what was once considered a traditional schoolgirl style, nowadays it has become popular again thanks to clever styling. But trying to work out what shape someone’s ponytail will be has puzzled scientists and artists since Leonardo da Vinci.

In 2012, scientists from the University of Cambridge and University of Warwick developed the ponytail shape equation (PSE) to unravel some of the mysteries of the ponytail. Their model takes into account the gravity ($g$), the elasticity of the hair, the presence of random curliness of hair, and an outward swelling pressure that arises from collisions between the component hairs (which explains how a bundle of hair is swelled).

This equation can be used to find $R$, the radius of the ponytail, in terms of $s$, the arc length along the ponytail. The length at which gravity bends the hair is $l$, $L$ is the length of the ponytail, $P$ is the pressure due to the hairband, $A$ is the bending modulus, and $\rho$ is the hair’s density.

The Rapunzel number, $\text{Ra}$, of a ponytail is the ratio $L/l$. This dimensionless number determines the effect of gravity on hair. When $\text{Ra}<1$, the hair doesn’t bend much, leading to a thin, straight ponytail. When $\text{Ra}>1$, the hair bends strongly under gravity leading to a wide, bushy ponytail.

The relevance of this equation is that it could help in understanding the structure of materials made up of fiber and depicting hair realistically in animation and video games. But most importantly, if you want to look good at a party or a maths conference, simply calculate your Rapunzel number and pop on a hairband that exerts the correct pressure.


  1. RE Goldstein, PB Warren, RC Ball, Shape of a Ponytail and the Statistical Physics of Hair Fiber Bundles, Physical Review Letters, 108, 078101, (2012).
  2. RE Goldstein, (2016, September 11), Leonardo, Rapunzel and the Mathematics of Hair.