Prize crossnumber, Issue 15

Our original prize crossnumber is featured on pages 60 and 61 of Issue 15.


  • Each of the letters $r$, $s$, $t$, $u$, $v$, $w$, $x$, $y$, and $z$ represents a different integer between 1 and 9 (inclusive).
  • The clues are provided as normal, but the locations of black squares in the grid are not given. Instead, you are given the position of every zero in the completed crossnumber.
  • The completed grid has order 4 rotational symmetry.
  • The starting squares for the clues are numbered from 1 to 60 in the usual order: starting in the top left corner and proceeding left to right for each row.
  • Solvers may wish to use the OEIS, Wikipedia, Python, an abacus, 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 all the digits in the row marked by arrows using this form by 30 September 2022. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by a November 2022.

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Dear Dirichlet, Issue 15

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

Dear Dirichlet,

We’re getting married in Shetland next month and are having a local band come to play some folk tunes at the reception. Doubtless the weather will be good so we’ve hired a remote, unpowered, outdoor barn for the occasion. For extravagant reasons, some of our guests will only dance with certain people, and not with others. But I’ve only just got the breakfast seating sorted and now this…! Can you help with this dancing dilemma?

— Martha Tracy, on a boat

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Cryptic crossword, Issue 15

Cryptic #3, set by Logisca: Download as a PDF or read on!

Across clues

  • 1. By morning, Somerville will be a fields medallist. (6)
  • 4. It’s Mats Vermeeren’s favourite mathematician that is debased. Not her! (7)
  • 8. In a direction somewhat antipodal to 17. (5)
  • 9. Elects to decrease one degree by an order of magnitude (ie a tenth). (7)
  • 10. Established time zone. (1,1,1)
  • 12. A mark by a learner is situated on an axis. (4)
  • 13. Magazine editors at heart regularly should (with slack) organise. (9)
  • 15. Imaginary degree from learned society. (1,1,1)
  • 16. The value approached is an extremity by the sounds of it. (3)
  • 20. Two yields level best… (4,5)
  • 22. … a single time produces no change nowadays. (4)
  • 24. Starts of lemmas, examples, and theorems, and the beginnings of some proofs. (3)
  • 26. European adopting one French mathematician. (7)
  • 27. See 27. (5)
  • 28. Bronze man is touching but not crossing. (7)
  • 29. Easing treatment provided by the witch. (6)

Down clues

  • 1. Men lead current top three in science, causing results of tension. (7)
  • 2. Revolutionary hands around a mixed acid. (7)
  • 3. Race around a large area. (4)
  • 5. Campaign division, say. (9)
  • 6. Helium-59? According to Sophie Bleau, it’s a circle! (5)
  • 7. Outcome of Ulster riots. (6)
  • 11. This could be you? (leftmost of all of the answers ultimately). (3)
  • 12. A computing pioneer whichever way you look. (3)
  • 14. A measurement of American money. First nickels, then assorted lead-free coins (9)
  • 16. Algebra of untruth? (3)
  • 17. One initially Cartesian plane transforms to historic polar opposite of 8. (7)
  • 18. Othello’s ancestor is about the fifth king in Spain? Yes! (7)
  • 19. Nothing new, argued uninitiated. (6)
  • 21. About 45 inches?! Sounds like hell. (3)
  • 23. It’s about time to fix back gate by church. (5)
  • 25. Odd terming is used when discussing triangles? (4)

The big argument: Are whiteboards better than blackboards?

Yes: whiteboards every time, argues Ellen Jolley

I do see the irony of writing an anti-blackboard piece in a magazine called Chalkdust, but I have to be honest: writing on a blackboard is simply not practical, and sadly the dust in particular is a key culprit. Sure, we all feel smart and fancy writing our equations on blackboards—but what about afterwards when you look down to find your treasured Chalkdust T-shirt caked in its namesake? Not so fancy now are we? The problem doesn’t stop at your clothes either: your hands, all the surfaces in the room, the board itself, the insides of your students’ lungs… Expect a layer of residue covering each.

We can’t blame the dust for everything: of course we cannot forget that blackboards are also the source of the noise which is universally accepted to be the worst ever. Even keeping your fingernails out of the fray, the chalk itself is constantly squeaking. It is also harder and slower to write with chalk than with a marker, it’s a much bigger pain to rub off, and it constantly snaps or shrinks.

Old-fashioned is generally not better (hence how it came to be old-fashioned), and mathematicians’ insistence on using this ridiculous medium does nothing but harm our carefully crafted reputation as cool-headed, logical people. Time to leave chalkdust where it belongs: on the cover of this magazine and nowhere else.

No: blackboards are our raison d’être, argues Sophie Maclean

Without blackboards being associated with maths, we wouldn’t associate chalk with maths.

Without chalk being associated with maths, we wouldn’t associate chalkdust with maths.

Without chalkdust being associated with maths, we wouldn’t have Chalkdust.

Ergo, without blackboards, we wouldn’t have Chalkdust. Need I say any more?


My favourite LaTeX package

Adam Townsend: cleveref

If you want cross-references in your LaTeX file (spoiler: you do), then you need to learn about cleveref.

Instead of filling your document with theorem \ref, section \ref and \eqref, you can just write \cref and cleveref will automatically write what kind of thing you’re referencing. It even does ranges!

And if you decide later that you want to make ‘theorem’ start with a capital letter (spoiler: you don’t), simply add some cleveref settings to your preamble and they’ll all change.

Madeleine Hall: booktabs

Second only to amsmath and graphicx, the package booktabs is my favourite due to its unbounded capacity to improve your tables in LaTeX.

No more \rule{0cm}{8pt} to get more space between your data and the horizontal line above. No more borders of uniform width. No more vertical lines!

Throw out \hline and say hello to \toprule, \midrule and \bottomrule, and give your table data the space it needs to truly shine.

David Sheard: tikz

Tikz is an amazingly versatile tool whose primary function is to draw 2D graphics in LaTeX. While a little fiddly to get used to at first, it ends up being a really fun package to use (most of the time, except when it is infuriating—but that’s LaTeX for you)!

As an indication of its utility, it is used on about half of the pages of our printed magazine, sometimes in surprisingly hidden ways.

Matthew Scroggs: realhats

Usually, the command \hat puts a circumflex above a letter. When writing a paper, this can be confusing as hats can be used to denote many different things.

Realhats can fix this problem by allowing you to put a huge range of hats over your letters.
Letters with hats on: an example of the realhats LaTeX package


Book of the Year 2021

A few weeks ago, we announced the 7 book shortlist for the 2021 Chalkdust Book of the Year. We award two prizes: the Chalkdust Book of the Year (as chosen by our editors), and the Chalkdust Readers’ Choice (as voted for by our readers).

Chalkdust Book of the Year 2021

We found picking a winner of this award really difficult, as we’d picked many great books for the shortlist. And the winner is:

How to Think about Abstract Algebra

Lara Alcock

This book (Amazon UK, Waterstones, Oxford University Press) is a book that tells you how to think about abstract algebra.

You can read our full review of How to Think about Abstract Algebra here.

Chalkdust Readers’ Choice 2021

As well as picking our favourite from the shortlist, we held a vote for our readers to pick their favourite. The runaway winner of this poll was:

Maths Tricks to Blow Your Mind

Kyle D Evans

This book (Amazon UK, Waterstones) is his book all about viral maths problems.

You can read our full review of Maths Tricks to Blow Your Mind here.


Top Ten: Matrices

This issue, Top Ten features the top ten waves! Then vote here for your favourite items of mathematical clothing for issue 15!

At 10, it’s The Matrix Revolutions.

At 9, it’s The Beatles with And Your Bird Can Singular matrix.

At 8, it’s Huey Lewis and the News with Hip to Be Non-square Matrix.

At 7, it’s a surprise new release after 18 years of silence: The Matrix Resurrections.

At 6, it’s Adjacency Matrix by The Carpenters.

At 5, it’s Ruckus in B (matrix of) minor(s) by Wu-Tang Clan.

At 4, it’s Identity Matrix by Kraftwerk.

At 3, it’s what you get if you press F5 while playing this issue’s number one: The Matrix Reloaded.

At 2 this issue, it’s Hit Me With Your Rhythm Stick by Ian Dury and the Block Diagonal Matrix Heads.

Returning to the top of the charts this issue, 23 years after its original release, it’s The Matrix.


What’s hot and what’s not, Issue 15

Maths is a fickle world. Stay à la mode with our guide to the latest trends.

HOT NFT hexagons

Show your Twitter followers you’re an idiot by investing in a $3/month monkey

NOT NFT shapes of constant width

Rolls down non-fungible coin slots, and comes free with every non-fungible copy of Chalkdust

NOT Celebrating mathematical days

Twosday Tuesday! Pi Day! Lots of numerical fun to be had this year.

HOT Doing maths on these days

Who needs pi to more than 11 decimal places anyway?

HOT Researchfish

‘Whilst we thankfully live in a society where we can freely express our opinions’, these guys will spam your inbox then email your funders when you say on Twitter they have a silly name

NOT Researching fish

Some pretty cool agent-based swarming modelling possible

HOT Zoom chicken

Say goodbye and keep waving. Just… keep… waving… Behold! Your own private party!

NOT Host has ended the meeting


HOT Selling your puzzle to the NYT for $1m

Five pages of puzzles in this issue. One of them has to be worth something, right?

NOT Winning a Fields medal for $15k

That’s like, a tank and a half of petrol?

HOT Chalkdust launch parties

Pizza! Quizzes! OK maybe we’ve had enough quizzes to last two lifetimes.

NOT Downing Street parties

This issue currently racing Sue Gray for first to publish


Page 3 model: The potato Earth

William Thomson—better known as Lord Kelvin—was a British mathematician who became interested in the temperature of the Earth around the late 19th century. Global temperatures have risen since his time, but our ability to predict the future of our climate has not, despite the enormous threat global warming poses to our planet.

Baked potato

One of these is Earth…

Much like our current ignorance of the future, the 19th century was relatively ignorant of the past: Kelvin’s interest in global temperature was for a method to determine the age of the Earth.

His idea was roughly as follows: assume the Earth is like a baked potato left out on the kitchen table. While the skin has cooled completely, the hot interior mash is just waiting to scald the mouth of an unsuspecting potato thief. Over time, the parts of the potato near the surface will cool down more quickly, so the rate of cooling is proportional to the change in temperature gradient through the potato:$$\frac{\partial T}{\partial t} = \kappa_p \frac{\partial^2 T}{\partial x^2},$$where temperature $T(x,t)$ depends on time $t$ and distance to the surface $x$, and $\kappa_p$ is a constant depending upon the potato.

By assuming the skin is perfectly cool:$$T(0,t) = 0,$$and assuming a perfectly baked potato, ie the initial temperature is uniform:$$\frac{\partial T}{\partial x}=0 \quad \text{everywhere at $t=0$,}$$Kelvin found an approximate solution to this equation, and was able to estimate the age of the Earth to around a million years. Of course, this is out by a factor of about a million, but was more accurate than many contemporary estimates.

The Earth

…the other is a baked potato

It is a common misconception that this model gives inaccurate estimates because Kelvin was unaware of the decay of radioactive materials throughout the Earth’s crust, but these effects are relatively minor.

A more sensible correction has to do with thermal convection currents in the Earth’s mantle, but anyone who thinks either of these happen in a potato is clearly baked.