# Dear Dirichlet, Issue 17

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

### Dear Dirichlet,

My neighbour’s dog must be playing pranks on me. Every morning I leave the house to find more garden ornaments sitting cheerfully outside my front door. At first it was just 1 or 2, or groups of three, but now they’re forming little football teams, in 1-4-6-4-1 formation. I fear for the hyacinths.

— Melody Mottlebrook, King’s Lynn

# Prize crossnumber, Issue 17

Our original prize crossnumber is featured on pages 58 and 59 of Issue 17.

### Rules

• Solvers may wish to use the OEIS, Wikipedia, Python, a slide rule, 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 23 September 2023. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 November 2023.

# What’s hot and what’s not, Issue 17

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

### HOT That new tile

In cinemas everywhere, on every website you visit and in every pop science thing for the foreseeable future. But who doesn’t love Einstein?

### NOT Penrose tiling

Yesterday’s aperiodicity. Knock down any buildings that feature it and start again.

### NOT Expecting optimal things to be beautiful

The best-known packing of 17 squares into a bigger square taught us to let it go

### HOT Getting it out the door

The universe is telling us: perfect is the enemy of done

### HOT Going out on strike

A great opportunity to bond with your colleagues’ dogs

### NOT Coming back to the same pay offer you rejected

And you think the dog might have scratched your jeans

### HOT Pythagoras’ field of beans

Issue 17 is full of sexy numbers, interviews with theorems and ‘LaTeX: expectation vs reality’

### NOT Kevin Costner’s Field of Dreams

Replace your corn fields with baseball to harvest the souls of the dead? Ludicrous!

### HOT Writing sonnets about maths & physics

’Shall I compare thee to a planetary orbit…?’

(either of them)

# Top Ten: Mathematical Monties

This issue, Top Ten features the top ten mathematical Monties! Then, vote here for your favourite mathematical games for issue 18!

At 10, he’s not a mathematician, but he definitely is a Monty: it’s Monty Don.

At 9, it’s mathematician, philosopher, and astronomer Guidobaldo del Monte (1545-1607).

At 8, it’s no-one’s favourite con trick: a three card Monte.

At 7, it’s the best Count never to appear on Sesame Street: The Count of Monte Cristo.

At 6, it’s the perfect breakfast to prepare you for falling asleep when you should be doing maths: The Full Monty.

At 5, it’s the Faculty of Natural Sciences and Mathematics, University of Montenegro.

At 4, it’s American statistician Irene Montie (1921–2018).

At 3, it’s the numerical method based on repeated random trials: the Monte Carlo method.

At 2, it’s the world’s nicest programming language: Python.

Topping the pops this issue, it’s the Monty most likely to give you a goat: a Monty Hall.

# Page 3 model: To betray or not to betray?

If you’re a fan of crime dramas you will have come across the following scene: the police have arrested two partners in crime, but are lacking evidence. They need a confession. The police try to turn them against each other: they tell the suspects that if they confess and give away their partner they can make a deal to reduce their sentence.

In the show, one of the suspects eventually confesses and takes the deal but is this how it goes in real life? This may seem more like a question for psychologists but if we assume that both suspects are completely rational, we can use a branch of maths called game theory to answer this question.

Consider a situation where both suspect A and suspect B can be arrested for a smaller crime, where they will serve 1 year in prison each (pay-off: −1 years). If they both confess to the bigger crime, they will serve 5 years each (payoff: −5). However if A confesses and B doesn’t then A goes free (payoff: 0) while B gets a 20 year sentence (payoff: −20). The same deal is presented if B confesses and A doesn’t. This is an example of a prisoner’s dilemma. Drawing a table of all possible actions and their payoffs in the form (payoff of A’s action, payoff of B’s action), we get:

suspect B
confess don’t confess
suspect A confess (−5, −5) (0, −20)
don’t confess (−20, 0) (−1, −1)

Looking from suspect A’s perspective, no matter what B does, A will get a higher payoff if they confess (−5 > −20 if B confesses and 0 > −1 if B doesn’t confess). So we say that not confessing is strictly dominated by confessing. This means that rationally A will always confess. Knowing that A is rational, B will have to either choose to confess and get a payoff of −5 or not confess and get payoff of −20. Therefore B will also choose to confess. We get the same results if we start from suspect B’s perspective. Therefore the only rational solution in this case is for both to confess.

Unfortunate for them but lucky for the detectives: another case closed!

# Book of the Year 2022

A few weeks ago, we announced the 8 book shortlist for the 2022 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 2022

We found picking a winner of this award even more difficult than usual, and ended up concluding that we would just have to declare two joint winners. These are:

## The Irrational Diary of Clara Valentine

#### Coralie Colmez

This book (Amazon UK, Waterstones, Bookshop.org) is her young adult novel about a maths-loving teenage girl.

## Math Games with Bad Drawings

#### Ben Orlin

This book (Amazon UK, Waterstones) is a book containing 75¼ mathematical games, with accompanying illustrations.

You can also read our full reviews of Math Games with Bad Drawings and The Irrational Diary of Clara Valentine.

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:

## An Introduction to the Math of Voting Methods

#### Brendan W Sullivan

This book (Amazon UK) is a book that that introduces readers to a variety of voting and ranking methods and encourages them to think critically about how to select the winner of an election, and why the winner may well change if the voting method changes even if the individual voters do not change their mind.

You can read our full review of An Introduction to the Math of Voting Methods here.

# Top Ten vote issue 17

What is the best mathematical game?

View Results

# Cryptic crossword, Issue 17

Once the puzzle is completed, 13 of the navy dots (each with a different number) will be in squares containing the letter E. These dots can be joined with straight lines in order (like a dot-to-dot) with an extra line connecting dot 13 to dot 1 to reveal a topical shape.

### Across clues

• 1. This measures state margin with hesitation. (7)
• 4. An even weightless weight. (5)
• 7. Origin’s rooted along diameter which is two of these. (5)
• 8. Domed shape formed by two union symbols at the entrances—left and right. (7)
• 9. Infinitely sum integers eg rank. (9)
• 11. Here Rory regularly counts. (1,1,1)
• 12. That is a slow browser. (1,1)
• 13. One in France can make things negative. (2)
• 14. A Chalkdust puzzle editor, whichever way you look at it. (1,1)
• 15. 3 on page 61, and 1 on your computer. (7)
• 17. Euler’s numbers and letters. (2)
• 18. Hesitation to invert friction. (2)
• 19. Logo making leads to time and money. (1,1)
• 20. Hill tops between eagles’ nests. (3)
• 21. Legion without Dracula? (9)
• 23. Aside from function by bottomless toilets? (7)
• 24. A mathematician on the radio of a tanker. (5)
• 25. Instrument for measuring part of some terabytes. (5)
• 26. German uses roughly 17 estimates. (7)

### Down clues

• 1. Blimey soldier. It’s a dog! (5)
• 2. With time, yield produces a black stone. (6)
• 3. A single hyperbolic paraboloid? (7)
• 4. Evacuate ancient Eastern politician by final Orient Express launch. (5)
• 5. A long litre fermented is only worth about 80% as much in the US. (6)
• 6. Forces hard into dependencies. (7)
• 8. Spooner’s nerd only read aloud a maths magazine. (9)
• 10. And I’m game! (3)
• 13. 14 down but not down. (2)
• 14. Test a vehicle—reversing’s a charm! (7)
• 15. 500 + 1000 = a unit? (2)
• 16. Result samples devout comedy. (7)
• 22. For example, $\sqrt{2}+\sqrt{3}=\text{irrational} – \operatorname{ab\,s}$ (5)