Prize crossnumber, Issue 14

Our original prize crossnumber is featured on pages 60 and 61 of Issue 14.


  • The down clues are provided as normal. The across clues are provided in alphabetical order without their clue numbers or lengths.
  • Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. Solvers may wish to use the OEIS, Wikipedia, Python, a book of log tables, etc to (for example) obtain a list of cube numbers, but no programming should be necessary to solve the puzzle. As usual, no numbers begin with 0.
  • One randomly selected correct answer will win a £100 Maths Gear goody bag, including non-transitive dice, a Festival of the Spoken Nerd DVD, and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping.
  • To enter, submit the sum of the across entries using this form by 14 March 2022. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 May 2022.

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Dear Dirichlet, Issue 14

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

Dear Dirichlet,

I’m not much of a football fan but I enjoyed the Euros over the summer. I’d like to learn a bit more about the sport and maybe support a club or two. Any suggestions?

— Steve Orgizovic, Unknown, possibly kidnapped

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Cryptic crossword, Issue 14

Cryptic #2, set by Humbug:

Across clues

  • 1. $\int2x\,\mathrm{d}x-x^2$, $6-6$, $6-10$, $1 – \text{O}$, $a\div h$. (6)
  • 4. Ape rambling by grand mountain. (4)
  • 7. How much is contained inside Far East?. (4)
  • 8. Fifty-one first, million second, six third, until eventually tending to this. (5)
  • 10. Limits of nonangle and torus can be folded up to make 3D shapes. (4)
  • 11. $\nabla\cdot(\text{kinetic energy}-y+t)$ is limitless. (9)
  • 14. Knight on the telephone regularly called after band. (6)
  • 15. Ball concealed by cusp, he realises. (6)
  • 17. Romeo, for example, coming to grips with TeX archive, initially loads editing box. (9)
  • 20. Boundary of zero-free area contains trivial origin‽ (4)
  • 22. A principle of astronomy leads people home again. (5)
  • 24. Contentless sudoku reworded to become irrational. (4)
  • 25. Head left, alternatively split in two. (4)
  • 26. Carla’s confused without direction. (6)

Down clues

  • 1. Disoriented duck halts magazine. (9)
  • 2. How quickly deep end fills up. (5)
  • 3. 50 and 5 dividing $3\times2.718$, adding $n$ makes a prime. (6)
  • 4. Endless DPhil leads to a solution of $x^2-x-1=0$. (3)
  • 5. Quadrilateral built from potassium, iodine and tellurium. (4)
  • 6. $x$, for example, starts after eleven seconds. (4)
  • 8. Function that can be split into smaller pieces using axes. (3)
  • 9. For example, $[0, 5)\in T$ precedes perturbed real around five?. (8)
  • 12. Calculates the area of misshapen triangle. (8)
  • 13. Dreamer in disarray produces rest. (9)
  • 16. Timeless tangles might be sharp?. (6)
  • 18. Third part of ratio mnemonic initially takes odd approach. (3)
  • 19. Is a bit of curve quality?. (5)
  • 20. Heading for Zambia Egypt result: nil-nil. (4)
  • 21. More complex than hyperbolic function. (4)
  • 23. Shared divisor ends much algebraic faff. (1,1,1)

Page 3 model: Woollen jumpers

As the weather turns from grey to cold and grey, it’s time to get out your favourite woolly jumper. Only, it’s a bit longer than you remember. Has someone has been stretching it out behind your back? Instead of jumping to conclusions, let’s try to understand how this wool-d happen.

Wool unstretched (A), and wool stretched (B)

Your jumper is made up of yarn, made from woollen fibres spun together. Let’s assume the fibres are in one of two states: $A$, unstretched, or $B$, completely stretched out.
The jumper as a whole is much harder to stretch with the fibres in state $B$ than in state $A$, because each individual fibre is already elongated.

As we start stretching the jumper, more and more fibres go from state $A$ to state $B$, and the rate at which it stretches slows down. Let’s model this stretching over time as
$$\frac{\mathrm{d}\ell}{\mathrm{d}t} = \alpha-\beta l,$$
where $t$ is time, $\ell$ is how much the jumper has stretched, and $\alpha$, $\beta$ are constants depending on the mechanical properties in states $A$ and $B$.

A jumper being stretched downwards

Assume that the jumper started off un-stretched (unless you got fleeced), so that $l(0)=0$. Then this equation has a unique solution, giving an explicit formula for how much your jumper has stretched:
$$\ell(t) = \frac{\alpha}{\beta}\left( 1- \frac{1}{\exp \left( \beta t \right)} \right).$$
On the bright side, even if someone has been wearing it, this shows your jumper won’t go on stretching forever: as $t$ gets very large, $\ell \approx \alpha/\beta$.

This approximate solution was reproduced by experimental evidence from the appropriately named Wool Textile Research Laboratory, just in case you thought this whole ‘two-state’ yarn was a bit of a stretch.


Top Ten: Waves

This issue, Top Ten features the top ten waves! Then vote here for your favourite matrix for issue 15!

At 10, it’s Walking on Punshine by Katrina and the Waves.
At 9, it’s the new single by Huey Lewis and the News: Hip to be a Square Wave.
At 8, and spending another week jumping up and down the charts: it’s a saw tooth wave.
At 7, Gravitational Pull vs. the Desire for an Aquatic Life by Stars of the Lid.
At 6, it’s Mex(ican wave) on the Beach by T-Spoon.
At 5, it’s Electromagnetic Wave My Fire by The Doors.
At 4, it’s Say Hello Wave (Function) Goodbye by Soft Cell (yes, they have songs other than Tainted Love).
At 3, it’s Bohemian Rhapsody by that wave the Queen does.
At 2 this issue, it’s Walking on Sunsine by Katrina and the Waves.
Topping the pops this week, it’s an electron behaving like a wave during the two slit experiment.

The big argument: Is dy/dx notation better than y’ or ẋ?

Yes: dy/dx all the way, argues Ellen Jolley

Do you take me for an engineer, sir? The only acceptable notation for a derivative is the original notation created by Leibniz: $\mathrm{d}y/\mathrm{d}x$. This is the only notation that demonstrates precisely what the derivative is: the limit of the change in $y$ over the change in $x$ as the change in $x$ tends to zero. I will not stand idly by as the integrity and beauty of the derivative is lost in a sea of dashes and dots.

Anyone who insists on such primitive notation clearly could not have ventured far in their study of calculus: any student of multivariable calculus can see plainly that these pathetic diacritics are simply not up to the task. A simple extension to Leibniz’s notation allows me to write with ease any order of partial derivative with respect to any number of variables I choose, meanwhile the dashing-dotting hooligans are left scrambling.

High schoolers can also reap the benefits of Leibniz’s original notation: they may initially be bemused by the concept of a fraction that is not a fraction—but as soon as they study integration by substitution, off and away they go, writing $\mathrm{d}u = 2x\,\mathrm{d}x$ and so on. What exactly are we to do with $u’$? Prime-root it? And I suppose $\dot{u}$’s reciprocal is $\begin{smallmatrix}\displaystyle u\\ \cdot \end{smallmatrix}$?

No: dy/dget in the bin, argues Sophie Maclean

As mathematicians, we love dealing with fundamental truths and as truths go, ‘humans are lazy’ is about as fundamental as it gets. Mathematicians, as a subset of humans (it’s true—I looked it up) must therefore also be lazy. And we can see empirical evidence of this—I once sat staring at a problem for an hour trying to work out which method of solving it required the least writing. I’m pretty confident the phrase ‘work smart, not hard’ was invented for mathematicians.

So then why would anyone ever write $\mathrm{d}y/\mathrm{d}x$?! The effort it takes when compared to $\dot{x}$ is vast. Furthermore this will occur repeatedly throughout a paper! If you thought writing $\mathrm{d}y/\mathrm{d}x$ out by hand is slow, wait until you try typing it in LaTeX. I should point out here that I’m not the one typesetting this argument, hence I have no qualms about repeatedly writing $\mathrm{d}y/\mathrm{d}x$. $\mathrm{d}y/\mathrm{d}x$, $\mathrm{d}y/\mathrm{d}x$, $\mathrm{d}y/mathrm\{d\}x$.

It’s also so much quicker to read $\dot{x}$ than $\mathrm{d}y/\mathrm{d}x$. I’m all about the marginal gains, but in this case the gains are on an astronomical scale! And then you get on to the environmental impact. Wasting paper space writing $\mathrm{d}y/\mathrm{d}x$, when $\dot{x}$ does exactly the same job, is frankly not justifiable. What would Greta say?


Top ten vote issue 14

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