Mathematics is an increasingly complex subject, and we are often taught it in an abstract manner. John Allen Paulos delves into the hidden mathematics within everyday life, and illustrates how it permeates everything from politics to pop culture – for example, how game show hosts use mathematics for puzzles like the classic Monty Hall problem.

The book is a collection of essays from Paulos’ ABC News column together with some original new content written for the book, on a huge range of topics from card shuffling and the butterfly effect to error correcting codes and COVID, and even the Bible code. As it’s a collection of separate columns, it doesn’t always flow fluently – I did find myself losing focus on some of the topics covered, particularly ones that didn’t interest me as much. This was mainly down to the content though – the writing style is extremely accessible and at times witty.

The book included some interesting puzzles and questions, which were challenging and engaging, and included solutions to each problem – very helpful for a Saturday night maths challenge! I even showed some to my friends, who at times were truly puzzled. I loved the idea of puzzles being a means of sneaking cleverly designed mathematical problems onto TV game shows. It goes to show maths is everywhere!

I enjoyed the sections on probability and logic as these are topics I’m particularly interested in. One chapter also explored the constant $e$, where it came from and where else it pops up – a very interesting read. It does deserve more attention, as π seems to be the main mathematical constant you hear about, and I appreciated seeing $e$ being explored in more depth.

This book would suit anyone who seeks to see a different side of mathematics – which we aren’t often taught in school – and how it manifests itself in politics and the world around us. That said, it would be better for someone with an A-level mathematics background, as some of the topics could be challenging for a less experienced reader.

It’s mostly enjoyable and has a good depth of knowledge, including questions to test your mind. While I didn’t find all of it completely engaging, there are definitely some points made in the book that I’ll refer back to in the future!

]]>π is an irrational number that is equal to 3.1415926535 (to 10 digits). Things could have been different, however, if Dr. Edward J. Goodwin succeeded in passing Indiana Bill No. 246. This bill would have completely changed π and mathematics as a whole.

In 1894, Dr. Goodwin, a physician who dabbled in mathematics, claimed to have solved some of the most complex problems in math. Among these was the problem of squaring the circle, which was proposed to be impossible by the French Academy in 1775. This is impossible due to the fact the area of a circle is $\pi \cdot r^2$, where $r$ is the radius, and the area of a square is $s^2$, where $s$ is the length of each side.

This was proven by Ferdinand von Lindemann in 1882, and is what makes squaring a circle impossible.

In order to square a circle, $\pi \cdot r^2$ must be equal to $s^2$. For example, if $r=1$, we would have $\pi \cdot 1^2 = s^2$, or $\pi = s^2$. This would mean that each side of the square is equal to the square root of π, and since π is transcendental, there’s no algebraic expression that could describe π.

Regardless, Goodwin claimed to have done it, and published his paper to *American Mathematical Monthly* in 1894. It was gibberish, and no amount of understanding in mathematics would make his work comprehensible. He claimed nine different values of π across his many works, with one claim going as far as $9.2376\ldots$, “the biggest overestimate of π in the history of mathematics” (*A History of Pi*). When his theories weren’t becoming popular, he decided to take them to the Indiana State Legislature on January 18, 1897.

Goodwin had convinced his state representative, Taylor I. Record, to introduce House Bill 246 (Indiana Bill No. 246). House Bill 246 would make Goodwin’s method of squaring the circle a part of Indiana law. However, those in the legislature either didn’t understand or didn’t even glance at the bill – and the House Committee on Canals decided to pass it. Dr. Goodwin’s ridiculous bill was now headed to the senate.

At the statehouse where the senate took up the bill was Professor Clarence Abiathar Waldo, a mathematics professor from New York. When Waldo heard what the bill was about, he was shocked to discover he was in the middle of a debate on a fundamental principle of mathematics. He decided to intervene and talk to the senators about the repercussions the bill would have on everything mathematics, and was able to stop the bill from passing the second chamber.

After Waldo’s intervention, it was clear to everyone that the people involved in the attempted passing of the bill, including Dr. Goodwin, were all wrong, and it was ridiculous to define mathematical truth by law.

]]>The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.

]]>Rational numbers, when written in decimal, either have a terminating string of digits, like $\frac{3}{8}=0.375$, or produce an infinite repeating string: one well-known example is $\frac{1}{7}=0.142857142857142857…$, and for a full list of reciprocals and their decimal strings, the Aperiodical’s own Christian Lawson-Perfect has built a website which generates a full list.

I’ve collected some interesting observations about the patterns generated by the cycles of recurring decimals, and in particular several relating to $\frac{1}{7}$.

For anyone interested, there’s a surprising connection between reptend primes and Fermat primes.

The recurring cycle for $\frac{1}{7}$, which is $142857$, has a length of $6$ digits – and when the length of the cycle produced from a fraction is one less than the denominator of the fraction, the cycle is a **maximum** recurring cycle. When the denominator is a prime number $>5$, the rational number will always produce a recurring cycle, and many primes produce a maximum cycle. Such primes are called **reptend primes** and the first fourteen are $7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167$ (OEIS A001913).

The decimal expansion of general fractions of the form $\frac{n}{7}$ also have the interesting property that it contains the same digits in the same cyclic pattern for any value of $n$ up to $6$, but with the start digit shifted along each time: $\frac{2}{7} = 0.285714285714285714…$, $\frac{3}{7} = 0.428571428571428571…$ and so on.

This is not unrelated to the fact that $142857$ is the first **cyclic number**, and the only one not beginning with zero. Cyclic numbers (OEIS A180340) are so named because when a cyclic number $n$ is multiplied by the numbers from $2$ to $n-1$, it contains the same digits in a different order.

$142857 × 2 = 285714$

$142857 × 4 = 571428$

$142857 × 5 = 714285$

$142857 × 6 = 857142$

If we multiply by multiples of $7$, an interesting pattern appears, which when an addition is applied, shows a further connection:

$142857 × 7 = 999999$

$142857 × 14 = 1999998 \rightarrow 1 + 999998 = 999999$

$142857 × 21 = 2999997 \rightarrow 2 + 999997 = 999999$

$142857 × 392 = 55999944 \rightarrow 55 + 999944 = 999999$, where $392$ is $56 × 7$

Multiplication by any number greater than $7$ but not a multiple of $7$ can also be seen to produce, via an addition, more cycles of $142857$:

$142857 × 9 = 1285713 \rightarrow 1 + 285713 = 285714$

$142857 × 37 = 5285709 \rightarrow 5 + 285709 = 285714$

$142857 × 127 = 18142839 \rightarrow 18 + 142839 = 142857$

$142857 × 3123 = 446142411 \rightarrow 446 + 142411 = 142857$

$142857$ is also a **Kaprekar number** (OEIS A006886) – when squared, the two halves of the number sum to the original number. Examples include $45^{2}=2025$ for which $20+25=45$, and $297^{2}=88209$ for which $88+209=297$. For our cyclic number, $142857^{2}=20408122449$ and $20408+122449 = 142857$.

Dattatreya Ramachandra Kaprekar (1905-1986) was an Indian mathematician who said of himself, “*a drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned”.*

Adding subsets of digits of cyclic numbers can also be seen to create interesting patterns:

\[14+28+57=99 \qquad 142+857=999 \qquad 1428+5714+2857=9999 \]

We can play the same trick with the second cyclic number $0588235294117647$: $94117647+05882352=99999999$; and using the third cyclic number $052631578947368421$ we have $947368421+052631578=999999999$.

We can also play with squares of subsets of the digits of cyclic numbers:

\[857^{2}-142^{2}=714285 \qquad 94117647^{2} – 5882352^{2}=8823529411764705 \]

For other prime denominators, we don’t always get cyclic patterns like this. The denominator $3$ gives two different cycles, $33333…$ and $66666…$ as does the denominator $13$ ($0769230769230769…$ and $1538461538461538…$). The denominator $11$ yields five different cycles. (For more, read Conway and Guy’s The Book of Numbers).

There is an extraordinary outcome for the set of proper fractions with prime denominator $19$. The maximum recurring cycle for $\frac{1}{19},\frac{2}{19},\frac{3}{19},…..\frac{18}{19}$, when laid out in tabular form, produces a magic square: its rows, columns and leading diagonals have the total $81$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The first row of the table is the recurring cycle for the rational $\frac{1}{19}$, and the second row is the cycle for the rational $\frac{2}{19}$. The digits for each subsequent row continue in a similar way leading to the final row, the cycle for the rational $\frac{18}{19}$.

Another nice pattern here is the remarkable fact that the cycle $052631578947368421$, the recurring section from $\frac{1}{19}$, can be generated from the the powers of $2$. Writing the powers of two in subsequent rows, starting on the right, and then adding the columns gives:

1 | |||||||||||||||||||||||||

2 | |||||||||||||||||||||||||

4 | |||||||||||||||||||||||||

8 | |||||||||||||||||||||||||

1 | 6 | ||||||||||||||||||||||||

3 | 2 | ||||||||||||||||||||||||

6 | 4 | ||||||||||||||||||||||||

1 | 2 | 8 | |||||||||||||||||||||||

2 | 5 | 6 | |||||||||||||||||||||||

5 | 1 | 2 | |||||||||||||||||||||||

1 | 0 | 2 | 4 | ||||||||||||||||||||||

2 | 0 | 4 | 8 | ||||||||||||||||||||||

4 | 0 | 9 | 6 | ||||||||||||||||||||||

8 | 1 | 9 | 2 | ||||||||||||||||||||||

1 | 6 | 3 | 8 | 4 | |||||||||||||||||||||

3 | 2 | 7 | 6 | 8 | |||||||||||||||||||||

6 | 5 | 5 | 3 | 6 | |||||||||||||||||||||

1 | 3 | 1 | 0 | 7 | 2 | ||||||||||||||||||||

2 | 6 | 2 | 1 | 4 | 4 | ||||||||||||||||||||

5 | 2 | 4 | 2 | 8 | 8 | ||||||||||||||||||||

. | . | . | . | . | 6 | ||||||||||||||||||||

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

$142857$, the recurring cycle for $\frac{1}{7}$, similarly occurs in this sequence of infinite additions. It begins with $7$ at the top, and each subsequent row is obtained by multiplying by $5$ and moving the end of the number to the left one each time. The cycle $142857$ is produced by adding the columns generated.

7 | ||||||||||||||||||||

3 | 5 | |||||||||||||||||||

1 | 7 | 5 | ||||||||||||||||||

8 | 7 | 5 | ||||||||||||||||||

4 | 3 | 7 | 5 | |||||||||||||||||

2 | 1 | 8 | 7 | 5 | ||||||||||||||||

1 | 0 | 9 | 3 | 7 | 5 | |||||||||||||||

5 | 4 | 6 | 8 | 7 | 5 | |||||||||||||||

2 | 7 | 3 | 4 | 3 | 7 | 5 | ||||||||||||||

1 | 3 | 6 | 7 | 1 | 8 | 7 | 5 | |||||||||||||

6 | 8 | 3 | 5 | 9 | 3 | 7 | 5 | |||||||||||||

3 | 4 | 1 | 7 | 9 | 6 | 8 | 7 | 5 | ||||||||||||

. | 0 | 8 | 9 | 8 | 4 | 3 | 7 | 5 | ||||||||||||

. | 4 | 9 | 2 | 1 | 8 | 7 | 5 | |||||||||||||

. | . | . | . | . | . | . | 7 | 1 | 4 | 2 | 8 | 5 | 7 | 1 | 4 | 2 | 8 | 5 | 7 |

I was surprised to discover in David Wells’ excellent Penguin Dictionary of Curious and Interesting Numbers that the points (1,4) (4,2) (2,8) (8,5) (5,7) and (7,1), formed by overlapping pairs of the digits in the $\frac{1}{7}$ cycle, lie on an ellipse, called the one-seventh ellipse:

Better still, we can concatenate pairs of digits from the cycle. The points created, (14, 28), (42, 85), (28, 57), (85, 71), (57, 14) and (71, 42), also lie on an ellipse, shown below:

Wells also stated that the points determined by the period of $\frac{1}{13}=0.076923076923….$ lie on a hyperbola. To find the equation of this conic, I contacted Professor Marc Chamberland – a co-author of the paper ‘*A Generalization of the One-Seventh Ellipse*‘. Marc sent this explanation in his reply:

Using the notation from our paper, we let $a=0, b=7, c=6$ and $S=9$. Theorem 1 implies that the points $(0,7), (7,6), (6,9), (9,2), (2,3)$ and $(3,0)$ all lie on a conic. Putting these points on the bivariate quadratic curve $Ax^{2} + Bxy + Cy^{2} +Dx + Ey + F = 0$, we can solve for the coefficients: $A = 141, B = 134, C=9, D = -1872, E = -684, F = 4347$.

Marc Chamberland

The conics were a huge and fascinating surprise but I don’t know if there is any conclusion or use for their connection to the cycles. However, there is a connection to the final part of the Pascal pentalogy as the paper A Generalization of the One-Seventh Ellipse makes use of Pascal’s Hexagrammum Mysticum Theorem.

If you’ve found this interesting, the following links are intended for any reader wishing to look further into the content:

- In particular, I recommend Tony Padilla’s Numberphile video explaining how the cyclic permutations are formed.
- MacTutor biography of Dattatreya Ramachandra Kaprekar, which includes some more number patterns
- You can read more about Cyclic numbers, Midy’s Theorem, Artin’s Constant and the general bivariate quadratic curve at Wolfram MathWorld.

As part of the 24 Hour Maths Game Show which took place at the end of October 2022, our own Christian Lawson-Perfect designed a maths/games crossover gameshow format to end them all – a mashup of hexagon-fighting TV quiz Blockbusters, and his own personal obsession: interesting mathematical factoids. Welcome to Blockbusters of Interesting Maths!

The premise of Blockbusters of Interesting Maths is simple. Start by collecting three maths communicators – in this case, me (**Katie Steckles**), Sheffield Hallam Uni maths lecturer and recreational maths fan **Alex Corner**, and mathematician and juggler **Colin Wright**. Colin is a self-described “torturer of adults and confuser of children”, but to clarify, he mostly does that using interesting maths. Alex teaches on the SHU Game Theory and Recreational Maths module with our own Peter Rowlett, and was prepared to have a good go at coming up with some interesting facts. I, on the other hand, have come across far too many interesting maths facts in my time, and can definitely half-remember most of them.

Between us, we’re pitted against Christian’s board of randomly chosen words – from Ogden’s Basic English, a collection of 850 common English words, from which he’s deliberately removed a chunk of the mathematical and scientific terminology. To make our way across the board, we pick a letter and find the word hiding behind, and are then charged with coming up with some kind of interesting maths fact relating to that word.

Christian’s judgement on whether our maths fact was interesting enough is final, and we’ve got to make an unbroken line from one edge of the board to another. If we fail to come up with a sufficiently interesting fact, or our fact is deemed too tangential to the word in question, that tile is blocked off.

Since we could never do anything the easy/conventional way, instead of a tessellation of hexagons, CLP’s gone for the Cairo pentagonal tiling, so each cell is only adjacent to five others instead of six. His web gadget, a version of which can still be found online for anyone to use (BYO interesting mathematicians), was deployed live on the Game Show to challenge the three of us, and the below is a blow-by-blow of what went down, with links to some of the things we talked about.

We’ve also included some additional facts from Christian, who is also a font of interesting maths facts and is making up for the fact that he didn’t get to play himself. Next time!

My initial instinct was to pass over to Colin, as he’s got a whole bit about calculating the distance to the moon using a pendulum, but instead he gave some interesting facts: the distance to the moon is pretty much exactly about 10 earth circumferences (~40 megametres), and it creates tides on opposite sides of the earth at the same time.

**Christian**: I can’t remember if anyone talked about all the different ways of counting a lunar month… I like the word sidereal and have no idea how many syllables it has.

We failed to come up with sufficient interesting maths for this – a bit of discussion about publishing results before someone else who’s working on them was deemed to be too depressing.

Alex talked about the work of Alan Turing on abiogenesis – mathematical models that can be used to describe patterns found on animal fur, including leopard spots and zebra stripes. Christian confirmed this was to do with reaction-diffusion models.

**Christian**: Back in 2008, a Simon Scarle published a paper connecting Turing’s work on reaction-diffusion to his other work on computability, through simulations of cardiac arrhythmia on the Xbox 360. I’ve never known what to do with this information. If you want to play with reaction-diffusion models yourself, there’s a good simulator called Ready, which we wrote about it in 2012.

I waffled briefly about the history of counting and the Ishango bone, which is an interesting historical artefact linked to early mathematical activity, and which it turns out I’d got mixed up with the Lebombo bone, which is an even older one.

Colin took this as a verb, and talked about predator-prey dynamics, particularly related to pursuit predation, including ambush and persistence behaviour in hunting. For each type of hunting, the animal has to weigh the probability of a successful catch against the amount of energy expended on the chase.

After a brief digression about which direction the real line points in (since we’d missed the opportunity to connect the board top-to-bottom, which most of us hadn’t realised was a thing), Alex couldn’t think of anything to say, so we lost this one.

After mentioning the mathematical use of the word, I managed to just about describe a particular maths problem this reminded me of that involved chasing something that’s swimming in a river (Christian mentioned this was covered in Dara Ó Briain’s School of Hard Sums, and it turns out there’s a writeup on Marcus Du Sautoy’s blog), and we then went on to another puzzle about a cat in a pond, which Ben Sparks has done a great video about.

**Christian**: Talking of interception reminded me of this fun paper describing a strategy for avoiding being intercepted while mapping an unfriendly subway system.

Colin covered a couple of topics – starting with control theory, which Colin compared to riding a unicycle. The trick is to keep the wheel under you, by (e.g) pedalling faster if you’re falling forwards, which can be understood by solving fairly straightforward differential equations – as unicycling robots often do.

He also talked about controlling a dog’s behaviour, and how rewarding good behaviour every time means the effect of training wears off more quickly, whereas rewarding it randomly some of the time means the effect lasts longer – this is related to spaced repetition as a learning technique.

Back over to Alex, who took electrical inspiration and used it as a chance to talk about capacitor laws. There were lots of nice relationships between different physical laws and it all got a bit physics, and as a result was rejected by Christian, so we lost this one.

I took the opportunity to talk about mathematical crystal structures, bond angles and 3D lattices (and got in a Kathleen Ollerenshaw mention). Christian also connected it to the structures of viruses, and mentioned Hamish Todd’s lovely videos.

Colin riffed on ratios in mixtures, from concrete to cake recipes, and then moved on to mixed techniques. Combinatorics, for example, uses a variety of different techniques you have to try in different combinations in order to solve a problem, and Colin explained how maths research, particularly in applied contexts, a mixture of techniques can be most powerful. Von Neumann showed that mixed strategies are always more effective in game theory!

After a brief digression about profit-loss models in economics, I jumped in with a mention of the version of internet protocols used in communication with objects in space, which Colin then ran with – talking about comms in trading (which relies on the speed of light to make sure transactions are instantaneous). A client-server model can be used, and in some contexts, equations from fluid mechanics are even used to describe how packets of information are moved around.

With that, we finally managed to satisfy Christian’s mathematical interestingness quotient and successfully connected the opposite sides of the board.

If you’d like to rewatch this or any other part of the 24 Hour Maths Game Show, you can find links to each segment on the website, and you can still donate to our charities by visiting 24hourmaths.com/donate.

]]>The leap second, referred to in this Independent article as a ‘devastating time quirk’, is finally being abolished. This has been covered in a bunch of places, mostly being quite rude about the leap second, including a writeup in the New York Times where it’s referred to as ‘a kludge, a bain, a pain in the little hand’ (£), and this Live Science article (‘pesky’). A committee at the International Bureau of Weights and Measures apparently nearly unanimously voted in support of Resolution D, meaning there won’t be any leap seconds from 2035 until at least 2135.

**Anti-maths news!** Princeton mathematician Rachel Greenfield (pictured left – photo by Dan Komoda/Institute for Advanced Study), working with Fields Medalist Terry Tao, has posted a disproof of the periodic tiling conjecture. A preprint titled ‘A counterexample to the periodic tiling conjecture‘ is now on the ArXiv, and if it’s correct, means that any finite subset of a lattice which tiles that lattice by translations, must tile it periodically. There’s a nice explanation in the Quanta writeup!

Meanwhile there’s been a new claimed proof of the 4-colour theorem, which is non-constructive (meaning it doesn’t rely on finding a colouring for every possible map, but proves the theorem generally). Some people have been skeptical about the proof, including in this statement from Noam Zeilberger, which links to a Mastodon discussion with John Carlos Baez. *(via Neil Calkin on Mastodon)*

Another claimed proof – this time of the sunflower conjecture. A k-sunflower is a family of k different sets with common pair-wise intersections, and the conjecture gives conditions for when such a thing must exist.

ArXiv has posted a framework for improving the accessibility of research papers on arXiv.org – their plan is to offer html as well as PDF versions of papers. *(via Deyan Ginev)*

Bright-trouser-wearer and mathematician Marcus Du Sautoy is offering a free OU online course, entitled ‘What we cannot know’. Find out how he manages to break the rules of reality by facilitating you knowing something that it’s by definition impossible to know, by signing up online for the 8-week course (which can also be accessed without signing in but then you don’t get a badge).

As part of their Elevating Mathematics video competition, the National Academies Board on Mathematical Sciences and Analytics (BMSA) invites early career professionals and students who use maths in their work to submit short video elevator speeches describing how their work in mathematics is important and relevant to our everyday lives, with a $1000 Prize for the best video.

And finally, in a rare instance of us linking to the Hollywood Reporter, Hannah Fry is to front a science and tech series for Bloomberg, entitled The Future With Hannah Fry. Sounds great! It’ll be available on Bloomberg’s Quicktake streaming service and will explore breakthroughs in artificial intelligence, crypto (not clear if -graphy or -currency), climate, chemistry and ethics.

]]>- Paul Glaister, Professor of Mathematics and Mathematics Education, University of Reading. Appointed CBE for services to education.
- Dan Abramson, headteacher of King’s College London Maths School. Appointed OBE for services to education.
- Kanti V. Mardia, Senior Research Professor, Leeds University. Appointed OBE for services to Statistical Science.
- Jeffrey Quaye, National Director of Education and Standards at Aspirations Academies Trust, PhD in Mathematics Education and Chartered Mathematics Teacher. Appointed OBE for services to education.
- Charlotte Francis, maths teacher and entrepreneur. Appointed Medallist of the Order of the British Empire for services to education.

Get the full list from gov.uk.

Updated 2/1 to add Dr. Jeffrey Quaye, HT The Mathematical Association on Twitter.

]]>This is a party trick that Katie Steckles showed me: you can fold a piece of paper and then make a single cut to produce a five-pointed star. I showed how to do it by following the instructions I’d been told, and then recreated the steps just starting from the insight that when you make the cut, all the edges of the shape need to be on top of each other.

Maybe you’ll show someone else how to do it during the Christmas holiday?

This doesn’t only work for stars: there’s a theorem that you can make any polygon by folding and a single cut. Erik Demaine has made a really good page about the theorem, with some examples to print out and links to research papers. Katie can cut out any letter of the alphabet on demand, which is impressive to witness!

]]>*We spoke to Coralie Colmez, mathematician and author of Math on Trial, about her genre-busting new Young Adult novel for mathematically minded teenagers: The Irrational Diary of Clara Valentine.*

*The Irrational Diary of Clara Valentine* is a fun novel aimed at readers aged 15-19. It’s got all the good things in it: a mystery, a sharp-witted narrator, an idiosyncratic best friend, a couple of charming potential boyfriends, a mostly-loveable family, and some maths!

I first had the idea for the book around 10 years ago. My mother (a mathematician) and I had just written a popular maths book called *Math on Trial*, which was really fun to do, but it made me realise that I wanted to find a way to write about maths which was closer to the things I like to read myself, and I mostly read fiction. Following the release of *Math on Trial*, I got the chance to talk about it at events for students, which I really enjoyed, so I decided to write for that age group.

I found it easy to decide which maths topics I wanted to cover – they are all my own favourites, and the things that made me love maths when I was a teenager! The topics are quite abstract and high-level, like countable infinity and logic – things that would normally only be introduced at university, even though they don’t require much prior knowledge and I think high-school students would really enjoy learning about them.

I was also interested in writing YA because I felt that, while there is a lot of great YA literature, none of it looked like my own experience of teenagerhood. Characters are either off dealing with a fantasy world, with major emotional trauma, living an extreme life (Euphoria-style), or on the contrary behaving in a totally PG way. There is a space and a need for all of these types of characters, but I wanted to try and write ones that felt more real, and could have been my friends and me.

Thank you, and I am glad you think so! Because the topics I wanted to cover are quite high-level, I had to find some creative ways to include them. I got a lot of well-meaning publishing professionals suggesting that I have Clara solve problems involving measuring angles, calculating the length of a rope and that kind of thing, but I was really set on sticking with more abstract concepts.

I definitely wanted to make sure there were a few different ways that the maths became part of the story: some of it happens via the characters in the book that know maths at a high level, but we also see moments like Clara teaching her little sister something, or Clara’s best friend learning a bit of mathematical history in her philosophy class. That’s what it’s like in my family (which is made up of 50% mathematicians, so not entirely representative): maths is just a part of normal life.

I also wanted to include what it’s like to think about maths, so I really liked getting in Clara’s mind when she is solving a question: how she approaches problems from different angles – how some of these angles sometimes don’t work at all – and how amazing it feels to crack a problem.

I really think that anyone could enjoy the book. As a novel, it’s a pretty exciting read, it’s funny, and hopefully I’ve managed to capture a bit of today’s really exciting generation of young people, who are so sharp, aware and witty.

When it comes to the maths, it’s written so that readers of different levels can take what they want. Someone who has quite a high level of maths might even try to solve some of the problems along with Clara, whereas someone who only has a basic interest in maths might simply enjoy the overall concepts that are introduced, like realising how different ‘Infinity’ is to just ‘A really big number’. In terms of the level of the maths, I would say that an 18-year-old reader who already has an interest in maths might already have heard of 2 or 3 of the 8 topics covered, but hopefully they’ll see even the ones they already know in a fresh way, with some new anecdotes attached! A couple of the topics are included in the A Level syllabus, though most aren’t.

The book is aimed at readers all of all genders, but as a woman maths graduate – the only one in my year at my college – with a mathematician mother, it’s really important to me to encourage more girls into maths and science. I hope that having an awesome (if I do say so myself) female narrator like Clara can help with that.

Finally, age-wise, I wouldn’t recommend the book to younger readers, because there are some themes that might be too old for them, and there is some sex (which is entirely age-appropriate, at least if you are French, and also very positive for ages 15+) I’ve had some very lovely comments from older readers though, so I’m going to say that there is no maximum age to enjoy the book!

When I finished my first draft of the book, I actually found an agent very quickly, and there was immediate interest from publishers when she sent out the manuscript. I’ll be honest, at the point when I was talking to three big publishers at once, I thought I was about to be famous! But in the end, none of them wanted to publish exactly the book I wanted to write. One wanted it for a younger audience, another wanted me to focus just on the mystery, the third wanted me to first write a novel with no maths, as they were nervous about Clara being a debut. There was a lot of talk about which ‘shelf’ my novel would fit in, and I realised I didn’t want to write a book that just fitted on one shelf, because that’s not what life looks like! Clara cares about maths, about her relationships, about solving a mystery, about politics, about doing well at school… just like we all do (well, apart from the mystery maybe).

You can get a copy on pretty much any online retailer, like Amazon – or if you are avoiding Amazon, it’s on Waterstones if you are in the UK and Bookshop.org if you’re in the US, for example. I’ve also put the PDF on my website, so anyone can read it for free. The book is pretty though, and I’m quite proud because I designed the cover myself, so I’d recommend getting a copy!

]]>Friends of the site Maths Gear have their usual selection of excellent gifts, including a new range of Maths Icons earrings (£14.99) including the Mandelbrot set, and a set of nested polygons. They also have a range of other jewellery and cufflinks which includes the wonderful mug/donut earrings (£17.89) and π cufflinks (£6.91).

There’s also some great Impossible Shape jewellery (from £3) – including Borromean Rings and Necker cubes – available from Earth Symbols, and there’s of course the classic Cofactor Pythagorean theorem earrings ($15 in 3D printed nylon).

Maths Gear also have a wide collection of different types of dice, as do scholastic suppliers Tarquin (including class sets and some individual items). We particularly like Maths Gear’s set of polyhedral dice in shapes they don’t usually come in (£10.97), and Tarquin’s set of blank rewriteable dice (£5.99), for when you want to make your own rules.

More generally, Tabletop Supply are a good go-to for many different types of dice, as well as replacement (or upgrade!) pieces for existing games, or games you’ve invented yourself.

Plenty of mathematically interesting board games come in travel versions which can fit into a small space – some of our favourites include number-based favourites Red 7, 6 Nimmt! and The Game, and if shapes is more your think we can also recommend travel Blokus and travel Qwirkle.

Oink Games also do some visually stunning and simple games with a mathsy vibe, and our favourites include Troika, Deep Sea Adventure and The Pyramid’s Deadline. We also love OK Play which is very compact.

Simple classic board games with a mathematical twist include Shut The Box, and you can’t go wrong with a deck of cards (this Math Stack variant (£10.49) from Maths Gear is pretty, but any will do!)

Site editor Christian recommends what he calls a Nobbly Wobbly, but tends to be sold under the name ‘woven bouncy ball’ or ‘rainbow spaghetti ball‘ (£5.95 for 5) depending on who you ask. The underlying geometry of the shape makes it mathematically interesting, but dogs and small children alike will have fun with it.

Happy Puzzle Co have IQ Minis which fit in your hand, and a range of other pocket puzzles. We also love Edmund Harriss’ Curvahedra (£11.93), which are available from Maths Gear. Paper-folding puzzle Manifold was a big hit with us a few years ago, and it looks like Manifold 2 is now available.

There’s also Rush Hour (£16.49), and we’ve had a recommendation for the STAX games from Huch’s range of puzzles, which comes in Cat, Dog and Sea versions. Don’t forget about the Rubik’s cube and other twisty puzzles too!

For a more refined mathematical palate, you can pick up some elegant vintage maths gifts from Present and Correct, including a 1970s desk abacus (£19.50), gorgeous metal protractors (£4), this ruler sticker tape (£3) and a classic geometry puzzle (£12). Or why not pick up a Golden Mean Compass (£14.99) from Grand Illusions?

If you have any suggestions of your own, feel free to include them in the comments below!

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