Crossnumber winners, issue 12

Hello everyone! As the 13th Chalkdust prize crossnumber has just been released, it’s time to announce the winners of the issue 12 crossnumber competition! Before we reveal the winners, here is the solution of the crossnumber.

6 7 2 9 9 9 7 4 1 3 2 4
4 2 2 1 9 9 9 8 0 1 5
2 1 4 1 4 9 1 1 3 7 6 2
8 4 1 2
1 5 1 4 4 7 3 3 3 9 4 5
5 4 4 4 2 4 3 3 3 9 4
1 4 8 7 3 8 3 3 3 3 3 9
4 3 9 2
1 4 1 4 1 1 2 6 4 1 0 1
4 7 9 2 1 1 9 2 2 2 0
4 4 1 9 6 1 7 5 9 1 2 4
3 9 5 4
7 2 3 3 3 3 1 5 1 3 6 3
5 1 1 3 3 1 5 6 9 3 0
7 3 5 3 3 3 4 9 5 3 6 0

The sum of the across clues was 19404.

There were 63 entries, 52 of which were correct. The randomly selected winners are:

  1. Robert Kerry, who wins a £100 Maths Gear goody bag,
  2. Nick Keith, who wins a Chalkdust T-shirt,
  3. Sarah Gross, who wins a Chalkdust T-shirt,
  4. Pamela Docherty, who wins a Chalkdust T-shirt.

Well done to Robert, Nick, Sarah, and Pamela, and thanks to everyone else who attempted the crossnumber. Hope you enjoy issue 13’s puzzle


The big argument: Is the Einstein summation convention worth it?

The Einstein summation convention is a way to write and manipulate vector equations in many dimensions. Simply put, when you see repeated indices, you sum over them, so $\sum_{i=1}^N a_i b_i$ is written $a_i b_i$ for example.

Yes: worth it, argues Ellen Jolley

This debate boils down to just one question: how much of your life do you spend doing tensor algebra? Those of us who undertake a positive amount of tensor algebra or vector calculus know that the goal is to be done with it as fast as possible! Try tensor algebra even five minutes without using the summation convention—I promise you will tire of constantly explaining “yes, the sum still starts from $1$, and yes, it still goes to $N$.”

You’ll scream, “All of them! I am summing over all indices! Obviously! Why’d I ever skip some??” If you’re confused how many you’ve got, use this simple guide: physicists use four; fluid dynamicists use three; and Italian plumbers use two. Wouldn’t it be nice to avoid saying this in every equation?

You may cry that it’s easier to make mistakes with the convention; but for applied mathematicians, the joy comes in speeding ahead to the answer by any means—time spent on accuracy and proof is time wasted. And as the great mathematician Bob Ross said: there are no mistakes, just happy little accidents!

No: not worth it, argues Sophie Maclean

Before writing this argument, I had to Google ‘summation convention’ which is all the evidence I need for why it’s just not worth it. I’ve learnt how to use the convention—multiple times! In fact, I’d say it’s something I’m able to use, yet I’m still not sure I know exactly what it is.

Some of our readers won’t have ever heard of it (which is one strike against it). Some have heard of it but won’t know much about it (another strike). But I guarantee none would be confident saying they can use it without making any errors (if you think you would be, you’re in denial).

We don’t even have need for the convention! We already have a suitable way to notate summation:
It’s taught to schoolkids. There is no ambiguity. And it’s so much less pretentious. Yes, the summation convention is fractionally faster to write out, but mathematicians are famed for being lazy and aloof—maybe dispensing with it is all we need to break that stereotype!


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On the cover: cellular automata

The game of life—invented by John Conway in 1970—is perhaps the most famous cellular automaton. Cellular automata consist of a regular grid of cells (usually squares) that are (usually, see the end of this article) either ‘on’ or ‘off’. From a given arrangement of cells, then the state of each cell in the next generation can be decided by following a set of simple rules. Surprisingly complex patterns can often arise from these simple rules.

While the game of life uses a two-dimensional grid of squares for each generation, the cellular automaton on the cover of this issue of Chalkdust is an elementary cellular automaton: it uses a one-dimensional row of squares for each generation. As each generation is a row, subsequent generations can be shown below previous ones.

Elementary cellular automata

An example rule

In an elementary cellular automaton, the state of each cell is decided by its state and the state of its two neighbours in the previous generation. An example such rule is shown to the right: in this rule, the a cell will be on in the next generation if it and its two neighbours are on–off–on in the current generation. A cellular automaton is defined by eight of these rules, as there are eight possible states of three cells.

In 1983, Stephen Wolfram proposed a system for naming elementary cellular automata. If on cells are 1 and off cells are 0, all the possible states of three cells can be written out (starting with 1,1,1 and ending 0,0,0). The states given to each middle cell in the next generation gives a sequence of eight ones and zeros, or an eight-digit binary number. Converting this binary number into decimal gives the name of the rule. For example, rule 102 is shown below.

Rule 102: so called because (0)1100110 is 102 in binary

Rule 102 is, in fact, the rule that created the pattern shown on the cover of this issue of Chalkdust. To create a pattern like this, first start with a row of squares randomly assigned to be on or off:

You can then work along the row, working out whether the cells in the next generation will be on or off. To fill in the end cells, we imagine that the row is surrounded by an infinite sea of zeros.

… and so on until you get the full second generation:

If you continue adding rows, and colour in some of the regions you create, you will eventually get something that looks like this:

It’s quite surprising that such simple rules can lead to such an intricate pattern. In some parts, you can see that the same pattern repeats over and over, but in other parts the pattern seems more chaotic.

The pattern gets a square wider each row. This is due to the state 001 being followed by 1: each new 1 from this rule will lead to another 1 that is one square further left.

But just when you think you’re getting used to the pattern of some small and some slightly larger triangles…Surprise! There’s this huge triangle that appears out of nowhere.

Other rules

Rule 102 is of course not the only rule that defines a cellular automaton: there are 256 different rules in total.

Some of these are particularly boring. For example, in rule 204 each generation is simply a copy of the previous generation. Rule 0 is a particularly dull one too, as after the first generation every cell will be in the off state.

Rule 204 is one of the most boring rules as each new cell is a copy of the cell directly above it.

Some other rules are more interesting. For example, rules 30 and 150 make interesting patterns.

100 rows of rule 30 starting with a row of 100 cells in a random state

100 rows of rule 150 starting with a row of 100 cells in a random state

If you want to have a go at creating your own cellular automaton picture, you can use this handy template. If you’d rather get a computer to do the colouring for you, you can download the Python code I wrote to create the pictures in this article and try some rules out.

There are also many ways that you can extend the ideas to create loads of different automata. For example, you could allow each cell to be in one of three states (‘on’, ‘off’, or ‘scorpion’) instead of the two we’ve been allowing. You could then choose a rule assigning one of the three states to each of the 27 possible configurations that three neighbouring three-state cells could be in. But there are 7,625,597,484,987 different automata you could make in this way, so don’t try to draw them all…


Prize crossnumber, Issue 13

Our original prize crossnumber is featured on pages 58 and 59 of Issue 13.


  • Each clue in this crossnumber contains two statements joined by a logical connective. If the connective is AND, then both the statements are true. If the connective is NAND, then at most one of the statements is true. If the connective is OR, then at least one of the statements is true. If the connective is NOR, then neither of the statements is true. If the connective is XOR, then exactly one of the statements is true. If the connective is XNOR, then either the statements are both true or they are both false.
  • 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 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, enter the sum of the across entries below by 18 September 2021. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 November 2021.

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Dear Dirichlet, Issue 13

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

Dear Dirichlet,

As a successful author on spies who are also fish, I’m looking to branch out a little. What with the number of streaming platforms, I’m hoping I can get a TV company to make my series of novels into a ten-episode drama. But it feels like a buyers’ market—how can I hook a producer? Let minnow!

— Micholas Herron, Oxford

Continue reading


Page 3 model: Cows

Have you ‘herd’? The world’s largest cow is over six feet tall and weighs more than 1.3 tonnes. Is a bigger cow possi-bull? Will the future contain infinitely large cows? The steaks have never been higher!

To answer this question, let’s take a look at the cow’s legs. If the main (meaty) bit of the cow has a volume $V$ and density $\rho$ then its weight is $\rho V g$. So each leg supports a load of about
\[N = \frac{\rho V g}{4}.\]
In pursuit of glory, let’s now make the length, height and width of the cow bigger by a factor $a$. The cow’s new volume is $a^3 V$ and so the load on each leg is $a^3N$: it grows cubically as $a$ increases.

Can the legs cope? If we model the legs as cylinders (since they already ‘lactose’…), we can use a 1757 result from the famous cow enthusiast Euler: if a cylinder has height $L$ and radius $r$, the maximum load it can support standing upright is
\[N_\text{max} = \frac{E \pi^3 r^4}{4 L^2}.\]
$E$ here is just a property of the material: its stiffness, or Young’s moo-dulus.

Cow with cylinders for legs

With our scaling, $L$ and $r$ are now $a$ times bigger. Our new maximum load is
\[\frac{E\pi^3a^4r^4}{4a^2 L^2} = a^2 N_\text{max}.\]

Uh oh… this only scales as $a^2$: quadratically.

So even though $N_\text{max}$ starts above $N$ (it has to, given that these cows exist!), there will come a maximum possible $a$, after which there will beef-ar too much cow and its legs will give way… an udder disaster.

This analysis tells us something really important about biology—that there is a natural maximum size for land mammals. But have we reached it for cows? Brody & Lardy’s 1000-page tome Bioenergetics and Growth from 1946 has all the de-tail you need. We’ll leave you to ruminate on the cow-culations.


Top Ten: Calculator buttons

This issue, Top Ten features the top ten calculator buttons! Then vote here on the waves for issue 14!

At 10, it’s Mambo No. 5 (A Little Bit Of…) by Lou Bega.
At 9, it’s All Apologies by Nirvana.
At 8, it’s Mambo No.5 (A Little Bit Of…) by Lou Bega.
At 7, it’s Up Allnight by Beck.
At 6, it’s Mambo No.5 (A Little Bit Of...) by Lou Bega.
At 5, it’s M+ambo No.5 (A Little Bit Of…) by Lou Bega.
At 4, it’s Mambo No.5 (A Little Bit 0f…) by Lou Bega.
At 3, it’s My Name = by Eminem.
At 2 this issue, it’s Thunderstruck by AC/DC.
At 1, it’s Mambo No.5 (A Little Bit Off…) by Lou Bega.