Horoscope, Issue 09

Mar 21 – Apr 19

Your research output doesn’t look so bright until a tall, dark, handsome stranger presents you with a proof of the Riemann hypothesis.
Apr 20 – May 20

The heavens are not in your favour, and people may try to take advantage of you. Don’t let them take you for a mug.
May 21 – Jun 20

Tomorrow you will wake up in a parallel universe which is identical to this one, except for the fact that no one ever invented the Banach–Tarski paradox. So you get to invent it! Good for you!

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Top Ten: Chalkdust regulars

This issue, Top Ten features the top ten Chalkdust regulars! Then vote here on the top ten issues of Chalkdust for issue 10!

At 10, it’s the page you probably didn’t use to find this page: the contents page.
At 9, it’s the big block of text on the page that no-one looks at: the editorial.
Did you know that the online vote is the reason that did you know made it to number 8?
At 7, but about to move to 14 as an infinite number of people have arrived, it’s Hilbert’s hotel: the game.
At 6, it’s the puzzles page. Can you work out why it’s so popular?
At 5, and causing a moderate amount of recursion, it’s top ten.
At 4, despite being deemed not hot, it’s what’s hot and what’s not.
Storming back into the top ten after not being seen since issue 03, it’s the horoscope.

Dear Dirichlet,

what is the second most popular Chalkdust regular?

Dirichlet says: No idea.

Topping the pops this issue, it’s the crossnumber.

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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Somewhere over the critical line

Maxamillion Polignac was a number. He was prime, and proud of it. One sleepy Sunday morning, with a cup of tea in hand, he opened the newspaper. The major headline shouted, prime club vandalised: composites blamed! Primes had been subject to prejudice for so long even though primes had founded the society. The strong dislike stemmed from primes being factors of composites. Maxamillion sighed with despair, but resigned, he continued reading.

He aimlessly scanned the newspaper until something caught his eye: the critical line: fact or fiction? The article speculated that the Critical Line was a fabled golden brick road that could lead to the Formula connecting all primes. It was supposedly hidden in the zeta landscape: an untouched land that defined the real and imaginary axes. The zeta landscape was complex and hard to navigate, and hence, the perfect place to hide the Formula. With the Formula, all the secrets underlying primes and how to find them would be revealed. Being a prime, Maxamillion did not have siblings, and living in a conflicted society, he felt alone, despite his many friends. But the Formula would give him a chance to find his twin prime who would make him feel complete.

With this idea on his mind, he went to his friend, another prime named Bernhard Oblong. Maxamillion said “I think we should try and find the Critical Line—we both get something: you’re a factorial prime, hence with the Formula, you could find out what your $n$ is; and I could find my twin prime!”

“Are you sure the Formula exists?” Bernhard asked.

“We have nothing to lose and everything to gain! Just like Pascal’s Wager.” Maxamillion reasoned.

“I hope you are right because I really want to know what $n$ is!” Bernhard agreed enthusiastically.

They embarked on their journey, oblivious to the dangers ahead. Starting in their city, the Number Line, they eventually reached a large building, the RSA bank in the outskirts of town. They noticed a sign with big black letters: no primes allowed.

“Let’s withdraw some money for the journey to the zeta landscape,” suggested Bernhard.

“Sure. But we better be careful,” replied Maxamillion.

A man with a baton stopped them. “Halt!” he said. “Can’t you read? no primes allowed! We don’t have primes coming this side of town.”

Maxamillion and Bernhard had no choice but to leave and try the next village. Another man, wearing a smug smile, saw the commotion through the glass doors of the bank.

“Hmm… primes. Interesting! Primes usually avoid banking with us because we use them to encrypt our credit cards. I know an opportunity when I see one. Let me see if I can capture them,” he thought to himself.

The slippery man phoned the most notorious prime-hunter in the zeta landscape—the Mersennary.

“Hello?” Mersennary crackled.

“I want you to capture two primes headed your way. Bring them back dead or alive. \$1,000,000 in credit cards,” the man ordered.

“It’s a deal!” Mersennary replied.

Their pockets empty, Maxamillion and Bernhard soon reached a small village. A sign hung over the entrance: \sign{this town is composite-free}.

“Wow. Those are some extreme views!” exclaimed Bernhard.

Not soon after, they found themselves surrounded by hundreds of primes who Maxamillion recognised immediately: Sophie Germain primes. They resisted any composite prejudice. They were well-built primes $(2p + 1)$ rebelling against the system and hoping to teach composites that primes are the building blocks of society, deserving equality.

“Who are you? Why are you here? Are you really primes or are you composite sympathisers?” The leader pelted questions faster than the duo could process.

“We are Maxamillion and Bernhard. We are trying to find the Critical Line. We are primes and certainly not composite sympathisers.” Maxamillion responded swiftly.

“Then you must be admitted into the UPS at once,” the leader proclaimed. “Follow Friedrich into the tent.” The two friends did as they were told.

As they were walking, Friedrich explained to them: “The UPS stands for United Prime Service. We are dedicated to protecting the rights of primes against the relentless prejudice of the composites. Join us. We are with the primes, we will continue to be so until the end.”

“Consider us members,” Maxamillion said. “Even though you are not as sturdy as us, we could use your help. Find the Formula and put an end to this!”

Before she sent them on their quest, the leader gave them a fascinating relic: $\zeta(s)$. “This is the zeta function,” the leader explained. “Use this once you reach the zeta landscape: it will help you navigate your way along the Critical Line.”

Progressively, the scenery morphed into a barren land: the zeta landscape. The two dimensions defined the real and imaginary axes. Using the relic, they navigated across the imaginary hills and the complex terrain. Even with hypothetical fog layering the land, they could clearly make out the glowing pathway. There it was—the Critical Line! “Whoa. It really is real. Really real.” Bernhard gasped. However, there was a dilemma ahead, for the Critical Line split into three paths.

Suddenly, a prime emerged from the fog. His sunken eyes added to the barren landscape. He whipped out his weapon, the square function. With it, the prime could square Maxamillion and Bernhard and turn them into composites. He advanced towards them, armed and dangerous. The duo trembled as beads of icy sweat trickled down their backs.

“I am the Mersennary. I am paid to hunt down primes like you,” he rasped.

Maxamillion tried to plead with him: “Why are you trying to break something that can’t be broken? We are all primes here. We have a rich history. Primes have been the dominant species in the whole of maths for hundreds of years. We ruled because we could not be broken down into other numbers. When we multiplied ourselves together, we created composites. Even though the composites have oppressed us, we remain strong and resistant. Primes will never be split. You are one of us, so are you a traitor?”

“Sorry. It is nothing personal, just business.” Mersennary responded coldly and inched closer.

In a desperate attempt, Maxamillion tried again: “Wait! You are in it for the money, right? War is not a steady business, and I am sure you would earn more at a new job. We want to get the Formula, which could help you learn more about yourself and other primes! You could use that to your financial advantage, eh?” Mersennary pondered and realised he had the wrong end of the number line. He decided to join them in the search.

Together they looked at the three new paths: $\operatorname{Li}(x)$, $\pi(x)$, and $x/\!\ln(x)$. $\operatorname{Li}(x)$ looked the most promising, because it went the highest, and it looked daunting enough to hide the Formula. $x/\!\ln(x)$ was very low, and it seemed like a place to start.

“Let’s go $x/\!\ln(x)$!” Mersennary said.

“No! Let’s go $\operatorname{Li}(x)$!” Bernhard replied.

Maxamillion urged: “Stop arguing! How about we compromise? Let us go explore the stairway $\pi(x)$. Maybe the Formula is hidden in the middle to stop people who aim too high or too low!”

They began climbing the never-ending stairs.

They were about to give up hope when they saw the Formula. It was $\pi(x)$. When Maxamillion touched it, it surrounded him with a blue light. Full of excitement, he asked the Formula to find his twin prime. But his enthusiasm didn’t last long as the formula would not give an answer. He sighed in desperation, but then he had an idea. He asked it the value of Bernhard. It answered 26951. Then he asked it the value of himself. It answered 26953.

“What?! We were twin primes all along?!” Maxamillion shouted.

“Wow!” Bernhard exclaimed. He then proclaimed: “With this, we can end prejudice! We could change the composites’ opinion of us by explaining all the secrets behind the primes and how intricate and beautiful we are!”

“We could also start a bank that serves all number-kind! Then I would have a steady source of income!” declared Mersennary excitedly.

As soon as they got back to Number Line, the trio started a bank: the Riemann bank. Soon it was booming and bought over the RSA bank. The first thing Maxamillion did as CEO was to demolish the sign saying, ‘no primes allowed‘. The law that barred the primes went down with the sign and they both crashed to the ground with a satisfying BANG!

The Formula and the relic were placed in Museum Polytechnique. The conflict between the two sets was finally resolved as the composites realised that the Formula revealed the complexity behind the primes. They realised that primes are so complex that they deserve to be treated better. Thus, the numerical landscape was changed forever!


Composites Numbers that can be written as the product of 2 or more primes.
Logarithmic integral, $\operatorname{Li}(x)$ An approximation of the number of primes until a certain given number, formulated by Gauss.
Mersenne primes Primes of the form $2^n – 1$.
Factorial prime Primes of the form $n! – 1$.
Natural logarithm A logarithm with base $\mathrm{e}$, not base 10.
Pascal’s wager A wager that states that if you believe in God and God does not exist, you have nothing to lose. If God does exist, you have everything to gain.
Primes Numbers that do not have any factors beside 1 and themselves.
Riemann hypothesis Riemann’s conjecture deals with the locations of the solution to Riemann zeta function. It is the holy grail of mathematics.
Riemann zeta function An infinite series used to investigate properties of prime numbers.
Sophie Germain primes Primes in the form $2p + 1$ where $p$ is a prime.
Twin primes $n$ and $(n + 2)$ are primes.

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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Dear Dirichlet, Issue 07

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

Dear Dirichlet,

I’ve recently had the good fortune of winning three pigs at the village fete. However, I’m not sure whether my triangular garden is big enough for them as well as my collection of metal, wooden and other deckchairs. The pigs are of substantial size and my tape measure is not long enough to measure the longest side of the garden. I’ve also heard that pigs are very intelligent and would like to hear suggestions for entertaining them.

— Pearl among swine, Lower Brailes

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Prize crossnumber, Issue 07

Our original prize crossnumber is featured on pages 52 and 53 of Issue 07.

Clarification: Added “non-zero” to clues 10A, 19D.
Clarification: For 20D, 0 is not a factor of any number, so the number contains no 0s.


  • 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. 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, with free UK shipping. Find out more at
  • To enter, submit the sum of the across clues via this form by 1 August 2018. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 19 August 2018.

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e to thee x

I’m e,
If you don’t know me
I live
Between 2 and 3.

This is my story.

I’m misunderstood.
Everyone thinks I’m just 2.71
But there is more to me
That they don’t see
I wish they would.

And I’m named after a letter
Which makes things worse!
The other numbers laugh at me for that too.

So I say to them—look at the things I can do,
I mean, surely I’m the only number
To have walked this earth
And expressed themselves in verse?

But that’s not cool, that’s not fashionable,
It’s all about being able to
Express yourself as something fractional, instead.

Otherwise they say there is something wrong with you
You’re crazy,

Natural numbers can count themselves
They can easily find their space
Because our place
On that line
Is defined
By our digits
And I don’t know all of mine.

But there was a time
When there were three of us who didn’t fit.
Pi also didn’t know,
Exactly where to stand, or sit,
And next was i,
She was in a dimension of her own!

I thought the three of us
Were meant to be,
We were all part of the same identity.

But then we grew up.

I started to see Pi from a new angle,
She had nice legs,
I loved that rest of her body was basically a rectangle.

We went on a couple of dates.

But then came Pi Day:
Three, Fourteen.
She let them approximate her!

And she became
An overnight sensation,
A household name,
One of those faces
Everyone knows.

And lost in that world of meaningless approximation
She wouldn’t let me take her to more than two places.

Time passed,
And even my old friend i and I grew apart,
The headiness of youth
Replaced by the steadiness
Of age,
I began to see what others had told me
And I’d refused to believe:
i was imaginary!

You might be wondering what about Tau?
She is twice the number Pi will ever be,
But Pi and Tau they are similar
And for me it was all a bit familiar,
Tau, she’s just too Pi.

So that’s it, back to lonely me.
It was hard
And I’m not a negative number.

But then
I met her,

She said ‘I’m $x$’,
I asked, ‘are you a multiplication sign?’
She laughed, ‘I get that all the time’,
‘No, I’m curly $x$,’
She said.

She’s curly $x$, she’s sort of curvy $x$,
But that’s not it,
I’m not one to make judgements
Based on digits or figures,
She’s different,
She’s fun.

And it’s true
I can’t always work her out
But I like that, too.
With her is where I always want to be,
I want her to be my unknown quantity!

So I wrote her a poem
‘e to thee $x$’,
(It was better than this).

She opened it, tentatively,
She read it, awkwardly,
Other numbers could hear!
Never have I wanted so much
To just

e to thee $x$, this poem I’d written,
This part of myself I’d given
Was supposed to feel just right,
But it was the opposite
It was the inverse of natural

Actually, that poem, didn’t happen.
I just dreamt that.
It’s the 21st Century
I sent her a Snapchat.

I sent it and then screamed inside
Until she read it, and she replied.

I won’t tell you what she said
But the bit that is etched in my head
Is the final line, a string of Xs.

The first was curly $x$,
That’s her name,
The rest were to say
She’d like to see me again.

And if you’re worried that what is essentially
A joke about numbers
Just did something unexpected to your heart,
Then shame on you!
Numbers can have feelings too.

So this story is about me and $x$
And the number she’s shown me I can be,
I’m e, I live between 2 and 3,
I don’t know where exactly
I don’t care!

With $x$, I see things from a different view,
I laugh in the face
Of the quite frankly ridiculous number queue.

I’m me, I’m e!