Prize crossnumber, Issue 18

Our original prize crossnumber is featured on pages 52 and 53 of Issue 18.


  • Solvers may wish to use the OEIS, Wikipedia, Python, send a telegram to your old maths teacher, 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 mean of all the digits in the row marked by arrows using this form by 14 April 2024. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 June 2024.

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News, Issue 18

Spectre sighted in time for Halloween

Dave, Chaim and the ghost of Judi Dench

Dave and Chaim. Composite: Chalkdust / Caroline Bonarde Ucci, CC BY 3.0

Back in March, David Smith, Joseph Samuel Myers, Craig S Kaplan and Chaim Goodman-Strauss published the aperiodic monotile that they had discovered: the hat.

The hat tile

Copies of this tile can fit together to cover the 2D plane in an aperiodic manner—the tiling never looks the same if is it shifted by any distance (except zero metres) in any direction. Importantly, this tile also cannot be tiled periodically.

Before the discovery of the hat, the best aperiodic tiles we knew about were the Penrose tiles—a pair of tiles that can only be tiled aperiodically.

There was, however, one issue with the hat: the aperiodic tiling that it made had to include reflections of the hat. This left an open question: does a tile exist that can only aperiodically tile the plane that doesn’t need reflections of itself?

In May, David, Joseph, Craig and Chaim answered this question with this aperiodic tile:

The spectre

They call it ‘the spectre’ as it has no reflection, is 2½ hours long, and has a cameo by Judi Dench.

Mathematicians win lottery, every time; brings happiness

Boffins from the University of Manchester have shown you need to buy only 27 tickets to guarantee winning a prize on the UK national lottery.

David Cushing and David Stewart say that among their 27 tickets, at least one will have two numbers in common with any of the possible 45 million different draws. Each ticket in the national lottery has six numbers between 1 and 59, and everyone who matches all six with those drawn shares the jackpot. Cushing and Stewart say we should settle for less—being happy to win any prize is good enough.

The Davids say there are no shortcuts: they’ve shown no set of 26 is enough. They set up the question in Prolog, a programming language that they describe as “temperamental, like a horse” to find the minimum number of tickets for different lotteries. But don’t go nuts—the smallest prize in the Lotto is a lucky dip ticket for the next draw. At £2 a ticket, you could easily end up winning less than you spent.

Since they published their findings, the Davids have been interviewed in newspapers, on the radio, and have even been the subject of YouTube videos. Their only disappointment? Nobody’s referred to them as “boffins” in print yet, and that they’re yet to win big on the lottery.

Chalkdust can help with one of these, at least!

Not another one! Ninth Dedekind number found

In April, mathematicians successfully determined the value of the ninth Dedekind number—32 years since the discovery of the last one.

How did they crack this mathematical puzzle? And what exactly are the Dedekind numbers? This issue has the answers!


Cryptic crossword, Issue 18

Cryptic #6, set by Humbug: Download as a PDF or read on!

Across clues

    1. Cat eating one hundred and one spiky plants. (5)
  • 5. He mixed with cop on 1 January 1970. (5)
  • 8,15,27. Ode to a fly. Quote:
        “I ride far in air, n’ end disjointed…. (8,12,8)
  • 10,15,27.     …a faltered pilot, in a quaint fire,
        involved the heat equation, for example.” (7,12,8)
  • 11. Say “$\pi$ is exactly three” before a crowd. Apparently Sophus made this up. (3,5)
  • 12. Assemble outside there, approximately 2.718m back. (4)
  • 15. see 8 or 10 (12)
  • 19. 10 and 500 look very happy. (2)
  • 20. $\displaystyle mc^2 \frac{v-u}{a} \frac{2(s-ut)}{t^2} =\eta$. (3)
  • 21. O/h! That’s evil! (3)
  • 23. $\displaystyle\frac{A}{r^2} = \frac{nRt}{V}\frac{V}{R}$. (2)
  • 24. Moper paid tax, incorrectly estimated. (12)
  • 26. Otherwise one Al* fan’s outside last. (4)
  • 27. see 8 or 10 (8)
  • 29. $2+50+50+1+1-\mathrm{e}=2^9\times5^9$. (7)
  • 31. Insect on my back’s scary and soothing. (8)
  • 34. A belt moving a desk. (5)
  • 35. Looks like Smee’s back. (5)

Down clues

  • 1. Magazine is built from carbon, hydrogen, aluminium, potassium… and fine dirt. (9)
  • 2. Charlie and Oscar’s company. (2)
  • 3. In that right triangle! (4)
  • 4. Self identification. (2)
  • 5. A long time in the rain. (3)
  • 6. DIY ramp rebuilt to make a polyhedron. (7)
  • 7. $x^2+\left(y-\sqrt[3]{x^2}\right)=1$ is in the article. (5)
  • 9. Afterword: napped awkwardly before 9. (8)
  • 13. Electronic arts, inc.\ energy and amps. (1,1)
  • 14. Even real hepcats choose by vote. (5)
  • 16. Otherwise one Al* fan’s outside last. (5)
  • 17. Power in next one confusingly makes power. (8)
  • 18. The Pope’s chosen ones? (9)
  • 22. Al* is after mixed spice to make distinctive. (7)
  • 25. Deep letters end with this! (1,1)
  • 26. $\displaystyle\exists\, {-s},e, x = \frac{2s}{v+u}$. (5)
  • 28. Tim has energy at, say, 4 o’clock. (4)
  • 30. I start out needing education. (3)
  • 32. Mythical city hiding in Brittany somewhere. (2)
  • 33. London and Kent without a sea. (1,1)

The zero knowledge proof

Chalkdust HQ has been infiltrated by a secret non-mathematician! So we may as well make them attempt some maths. This issue we asked our secret non-mathematician to…

Prove that you cannot square the circle!

Well firstly, I find it surprising to be asked to prove that I cannot square the circle. I can hardly calculate the area of a circle in the first place. I first encountered the theorem that the area of a square cannot equal that of a circle (or in the pithy words of my editor, one can’t “construct a square with the area of a given circle by using only a finite number of steps with a compass and straightedge”) in the Arctic Monkeys song Don’t Sit Down ‘Cause I’ve Moved Your Chair. The brilliant Alex Turner there points out the danger of trying to “fill in a circular hole with a peg that’s square”. I always knew the man was a genius, but I now realise his talents extend to geometry too.

Yet the question of why one of you cannot square the circle certainly does vex me… I thought mathematicians were supposed to be the smartest of the smart? Shouldn’t you have solved it by now? I recall that Thomas Hobbes—the famous political philosopher—got into hot water in 1665 for claiming that he could square the circle. This was not surprising from a man who was fond of saying that if he had read as many books as other people, he would be as stupid as they were (some mathematicians may relate). But mathematicians since then seem to have lost their ambition. 

I know, I know, you say it’s ‘impossible’. But take this advice from a former politics student: one should never let something as trifling as an impossibility get in the way of a good proof. It seems you are no strangers to this kind of thing, anyway. Do you really mean to tell me that imaginary numbers are ‘possible’? Put some of them to work on this squaring the circle business and you might meet with success. 

Sadly for you, I have got there first. To square the circle you just need an infinitely large circle and infinitely large square. There, boom, you have it. Both areas are infinitely large so it’s problem solved. “That’s cheating!” I hear you saying, “you can’t use infinities!” Oh really, can’t I? Then show me a circle that actually has one side. Circles are not real shapes. There is no such thing as a circle. A curve is simply lots of small straight lines put together. Any low-resolution circle on MS Paint will show you this fact. So, far from being a one-sided shape, the circle is in fact an infinitely-sided shape. This means that if I am cheating by using infinity, then the circle is already cheating by being an infinitely-sided shape. Or (as is much more likely since infinities do not exist) the circle does not exist, and the question simply becomes how do you square the chiliagon (just Googled this) or some other such polygon with many sides. And that, dear reader, is one for you to solve. I think you will agree that I have done more than enough.

I was asked to prove that I can’t square the circle, and I am afraid I have shown the opposite. Be grateful; there is now one fewer problem for you to solve. And so we find ourselves back where we started; I suppose we have come full square.


Top Ten: Mathematical games

This issue, Top Ten features the top ten mathematical games! Then, vote here for your favourite mathematical norms for issue 19!

Down FIZZ! places to BUZZ! in this issue’s top BUZZ!, it’s Fizz Buzz.

At 9, it’s the version of Asteroids for fans of maths:

At 8, it’s the greatest puzzle game on the PlayStation 1: Devil Dice.

At 7, it’s the mathematical game with the best music: Tetris.

At 6, it’s the game with the most angles snooker.

At 5, it’s Martin Gardner’s Mathematical Games column in Scientific American.

At 4, it’s the game that’s both older and better than chess: Go.

At 3, it’s everyone’s second favourite game that came with Windows XP: Minesweeper.

At 2, it’s the only impartial game: Nim.

Topping the pops this issue, it’s the Countdown numbers game.


Orbit crossnumber

This guest crossnumber appeared on page 33 of Issue 18.

Download Orbit as a PDF, or read on!

Orbit, set by Kwig:

This crossnumber includes radial and clockwise clues. Half the radial entries are entered from the circumference toward the centre; the other half are entered from the centre outwards. All radial entries are four digits long except 1r, which includes the centre cell. All clockwise entries are entered clockwise. Every entry is different and none begin with zero.

Radial clues

  • 1r. The sum of this number’s digits is a cube; the product of its digits is a fourth power; and it is a multiple of 26c. (5)
  • 2r. The sum of the squares of 2c & 23c. (4)
  • 3r. A multiple of 22c. (4)
  • 4r. The length of the hypotenuse of a right-angled triangle with shorter sides 7r and 19c. (4)
  • 5r. The sum of the squares of 20c & 25c. (4)
  • 6r. A prime. (4)
  • 7r. see 4r. (4)
  • 8r. 19c + a square.$ (4)
  • 9r. 11r + a triangle number. (4)
  • 10r. The product of three distinct primes. (4)
  • 11r. The sum of four consecutive primes. (4)
  • 12r. 14r + a square. (4)
  • 13r. Another entry squared. (4)
  • 14r. 2r – a square. (4)
  • 15r. 4 more or 4 less than 16r. (4)
  • 16r. A triangle number + 23c. (4)

Clockwise clues

  • 2c. A square pyramidal number*. (2)
  • 4c. A hexagonal number**. (2)
  • 6c. The product of three distinct primes. (4)
  • 10c. 16r + 24c. (4)
  • 14c. 4c + a cube. (2)
  • 16c. A palindrome. (2)
  • 17c. A multiple of 20c. (4)
  • 18c. The sum of this number’s digit is a square. (3)
  • 19c. see 4r. (4)
  • 20c. A factor of 3r. (2)
  • 21c. 6r – a cube. (3)
  • 22c. 2c + 14c.$ (2)
  • 23c. The product of three consecutive primes. (2)
  • 24c. An anagram of 2r. (4)
  • 25c. One more than 20c. (2)
  • 26c. A square. (2)

* The $n$th square pyramidal number is the sum of the first $n$ square numbers.
** Hexagonal numbers are the number of dots in the patterns in this sequence:


Dear Dirichlet, Issue 18

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

Dear Dirichlet,

I’m excited to try my hand at home automation. For compatibility with my iPhone, I thought I’d get one of Apple’s Alexa-type things. But now every time I try to turn my smart bulbs on with my voice, my phone starts talking back at me, then the Alexa starts talking, which sets my phone off, which sets the smart speaker off… I can’t get them to shut up! You seem like a modern man: how can I end this caco-phone-y?

— Megan Squirrel, London

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What’s hot and what’s not, Issue 18

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

HOT Finding 27 lottery tickets that guarantee a win

Don’t live a little! 1, 2, 3, 4, 5, 6; 9, 10, 11, 12, 13, 14; 18, 19, 20, 21, 26, 27; …
This time next week you’ll be rolling in free lucky dips.

NOT Buying the same lottery ticket 27 times

Having to split the jackpot with yourself? We choose to STEAL.

HOT Using maths to find the ultimate swear word

Surely the Italians have found an accompanying gesture by now.

NOT Taking mathematicians less seriously because they have a sense of humour

HOT Coming away from a conference with a signed aperiodic tile

The perfect souvenir and office decoration for the ages.

NOT Coming away from a conference with Covid

Sorry Matt – see you next time.

HOT Pictures of birds

Dippers! Penguins! Swallows! Especially great if they tesselate.

NOT Blackboard bold

Introducing ℂ: a magazine for the mathematically curious. Introducing 𝕋: where every little helps. Introducing 𝔽: they’re GRRRReat! Who needs graphic design anyway?

HOT Double-sided conference badges

NOT New University Challenge

Why is his chair so small?

Why do the titles look like QI?

Why don’t the names fit in the box?

NOT Old University Challenge

Forever the worst part of Quizzy Mondays.