I took the kids to a science fair recently, and they tried their hand at marbling with actual paint… and wet paper, which ripped before we’d left the venue. Fortunately, I was reminded that it’s possible to do marbling mathematically. And it’s invertible, so you can recover your original image!
Speaking of inverses, that’s today’s Mathober prompt! FractalKitty is running it again; it’s a prompt-a-day, make-what-you-like challenge. (Personally, I’m trying to write a song verse every day; I know Katie is trying to write a daily crossword clue. The possibilities are endless.)
Computing!
Following on from the “computers are magic” thing last week, I’ve stumbled on, but not checked out, Arithmazium, which seems to be an explanation of how computers deal with numbers. Or, from a brief glance, doughnuts.
Shapes!
Once upon a time, I wrote about Ailles’ Rectangle — if you inexplicably prefer Wikipedia to my blog, here’s your link. It’s a really neat way to figure out the trig values for 15-75-90 degree triangles, and — if you play about with it a bit, to prove all sorts of identities.
I’ll be speaking at this year’s Big MathsJam, but I promise I will not be visiting every cell on the border of Camelot. It’s barely a month away. Eek!
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
Double Maths First Thing is Colin’s weekly assortment of mathematical news. Or what time’s manacle.
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight through the medium of mathematics. It’s Wednesday morning and it’s time for Double Maths First Thing.
On magic
It’s probably a bit gauche to start with an article I wrote, but it’s certainly something that caught my eye: Rob Eastaway (all-round good egg and author of Much Ado About Numbers, available wherever good books are made available) sent me a page from the diary of an Elizabethan impresario describing a card trick. Here’s my description of how it works — with a bit of help from young Bill.
A different kind of magic goes on under the bonnet of your average computer. (Computers should definitely have bonnets. “Oooths, looks like your fan belt’s gone, that’s going to be expensive.”) When you input a number, the computer takes it in as a string of characters. How does that get turned into an Actual Number? It’s surprisingly complicated.
On beauty
You say you have found beauty In Euler’s identity It’s basic trigonometry It’s Pi Day, I’m in a huff
Even when it’s not Pi Day, I get in a huff about the framing of Euler’s identity as “the most beautiful equation”. Andrew Stacey articulates it a lot more clearly than I would, and with less swearing.
I’ll accept that dance can be beautiful (I have a cousin who’s a professional choreographer, and who has a very stern Disapproving Look, so I have to say that). Here’s a nice piece about different styles of dance notation; my only criticism is that they don’t make a joke about Scottish country dancing needing a Ceilidh table.
On stupidity and getting things wrong
Another article that’s had me nodding along and saying “YES!” is this from Math For Love: it makes the powerful point that feeling stupid is an important part of becoming smarter — and it’s an entirely different thing from being stupid.
One thing that always makes me feel stupid is how our experience of the world is pretty much limited to an incredibly narrow shell — a plane at 30,000 feet is 0.14% of an Earth radius up in the air. It turns out that GPS and route-trackers generally are just… not very good at elevation.
A couple of final things: I recently stumbled on Cyrille Rossant’s Awesome Maths List — I’m sure some of you know of resources that belong on there, and he seems receptive to pull requests.
I’ll end with something else about me: this year, I’m doing something I’ve never done before. I’m going into my kids’ school to run a lunchtime code-breaking club around the National Cipher Challenge for years 5 and 6. (The headteacher almost bit my hand off, it sounds like they’re studying Bletchley Park this term). I’ve done the challenge before, it’s just the wrangling young’uns that’s new.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website. You can also just reply to this email if there’s something I should be aware of.
In a dimly-lit tavern on the South Bank of the Thames, Philip Henslowe — builder and owner of the Rose Theatre — is celebrating the success of Shakespeare’s latest blockbuster, Henry VI Part I, among the cutthroats, actors and other lowlife of London. He spreads thirteen playing cards on a table in a circle. “Pick a card,” he grins. “Any card.”
Henslowe, one of Elizabethan theatre’s most important figures, kept a diary. It’s mainly the accounts of the theatre and records of loans, but among the administrivia are some gems — including the following card trick:
Dulwich College Archive MS VII f18v — with kind permission of the Governors of Dulwich College
My dogged team of researchers is looking into it, but there are very few documented card tricks from this era — and most of them are sleight-of-hand or forces. Tiago says this might be related to something written by Pacioli in 1478, and I understand there are Italian deck-stacking tricks from the first half of the 16th century. While it’s relatively unremarkable now, it seems quite sophisticated for its time.
Luckily, the trick was transcribed by W. W. Greg barely 300 years after having been scrawled out by Henslowe:
Now, I don’t know about you, but I’m not fluent in shorthand-infused streams-of-consciousness written in Early Modern English. Maths communication has evidently come a long way in the last 400 years. Here’s the best I can do as a more-or-less faithful translation:
Take 12 cards and the jack of clubs and lay them in a circle like a clock, all face down except for the jack. Put the jack at the bottom like on your watch [ed: I have never owned a watch with the jack of clubs at the bottom of it, but let’s roll with it], laid out like [the picture]. Then ask the volunteer what time they will get up and to keep it to themself.
Tell them to pick a card to count from [ed: It’s unclear to me whether the trickster or the volunteer picks the card — it doesn’t make a difference, so I’d let the volunteer do it]. Starting from this card and moving clockwise, they should count from their card up to 15 — so if they picked 7, they should count on eight cards.
Going around the circle, you count aloud clockwise while pointing at the cards, saying “15” on the first card clockwise from the jack, “16” on the second and so on up to 26. Tell the volunteer that whichever number you said when you pointed at their current card, they should count anticlockwise from their secret number up to that number.
When they flip the card they land on, it will be the number they first thought of.
I presume “a proved” is Early Middle English for “and everyone said WOW! That’s amazing.”
But it doesn’t work.
If you follow the instructions — which, like a game of Telephone that started centuries before the telephone was invented, have been written down from Henslowe’s memory, transcribed by an expert from unclear manuscript, and then translated into modern-day English by someone unqualified to do so. Hi! — you’ll find your “tada!” falls flat, because it’s not their secret number.
Let’s try it: I get up when I want, except on Wednesdays when I’m rudely awakened by the dustmen at 6am. And, rolling a 13-sided die¹ to decide where to start, I get card #3. I need to count on 9 clockwise from there (to make it up to 15), so I end up on card #12. That’s been given the number 26, so I need to count counterclockwise from my number (6) up to 26 — that is, 20 cards backwards. That takes me to 5.
¹ Yes, I do own a 13-sided die. Why do you ask?
Close, but no not-yet-introduced-to-England cigar.
It turns out that, whatever card you start from, and whichever number you pick, you’ll end up on the card immediately before your secret number. This suggests an easy fix: start your counting-aloud from 14 at card #1.
In case you want to do the trick yourself correctly, here are instructions for my version:
Tell your volunteer to think of a secret number from 1 to 12 (don’t let me control your mind!).
Tell them to pick a card (any card! don’t tell me what it is) and count clockwise from their chosen card, starting from their secret number and ending at 15. Have them tell you the card they land on.
Now tell them the numbers attached to the card: whatever “time” their card is on the clock face, give them a number 13 higher.
They must now move anticlockwise from their chosen card, starting from their secret number, ending on the number attached to the card.
Turn over the card they ended up on and say “abracadabra!”.
My glamorous assistant Bill goes through the trick with me. Better magicians than me — which is pretty much everyone — will have ideas about improving the patter and performance.
NOW it works. But why?
You know what else, apart from the later Shakespeare plays, telephones, and cigars, hadn’t arrived in Elizabethan London? I’ll tell you: modular arithmetic. At least, modular arithmetic as we know it — working with remainders goes back to at least Sun Zi in the third century CE, but Euler and Gauss’s formalisations of it were still 150 years away.
I don’t know what Henslowe’s mathematical background was — he was certainly competent at regular arithmetic — so I don’t know whether he understood why the trick worked, whether he came up with it himself, or anything about the history of it. All the same, I’m certain he wouldn’t have used the modulo function.
(In case you’re one of today’s lucky 10,000: modular arithmetic uses the remainder left over when you divide by a given number, like on a clock: 16:00 is the same as 4pm, and we’d say we were working “modulo 12” or “mod 12”, because we are lazy and modulo is far too long a word. The numbers 4 and 16 have the same remainder when you divide them by 12. In this problem, we’ll be working modulo 13.)
Let’s say you’ve picked secret number \(s\) and you decide to start from card #\(c\). You’re going to count on \( 15-s \) cards from there, so you end up at card #\( (c + 15 – s ) \). (We can think of card 14 as the same as card 1 and so on.)
The number I assign to it is 13 more than the card number. Modulo 13, that’s just the card number — but doing it this way ensures we don’t have to deal with negative numbers. (Negative numbers had probably reached England by this point, but I don’t imagine they were the kind of thing you’d want to have in a card trick.)
In any case, the volunteer is currently at card #\( (c +15 – s ) \) and has been given the target # \( (c +28 – s )\) to count to, starting at their secret number \( s \). That means they’re going to move \( (c +28 – 2s) \) cards back the way they came, starting at card #\( (c + 15 -s ) \). Moving backwards makes it a subtraction, so we work out \( (c + 15 – s ) – (c +28 – 2s ) \) to see that we end up on card #\( (s – 13)\).
And, because there are 13 cards, that’s the same as card #\(s\), which has the volunteer’s secret number written on it.
Boom.
One more twist, though
When I talked to young Bill about it, he asked a tremendous mathematical question: “would it work with a number other than 15?” The kid is ten years old, and already making me mutter “good GRIEF, where did that come from?” about three times a month.
The answer is… you don’t need it to be 15. In fact, the first half of the trick is mathematically irrelevant². You could ask them to spell out their secret number in a language of their choice, you could ask them to add their age to their best Parkrun time in minutes, you could ask them to spin a coin and pick the card it lands closest to. It doesn’t matter in the slightest, as long as they pick a card.
² That doesn’t mean it’s not an important part of the trick! I think it’s helpful to demonstrate how you want the final bit counted, and it misdirects the volunteer/audience into thinking there must be something clever going on.
If that’s card #\( C \), then they subtract \( (C + 13) – s \) from it — which again leaves you on card #\( (s – 13) \), which is card #\( s \).
Even knowing the maths behind it, I think this is still a pretty impressive trick. To someone frequenting a smoky Elizabethan tavern, it must have looked like, well, magic.
Thanks to Rob Eastaway for sending me the trick. His book on the maths of Shakespeare, Much Ado About Numbers, is available wherever good books etc.Thanks also to Paul O’Malley and Tiago Hirth for historical help, and to Calista Lucy and the Governors of Dulwich College for permission to reproduce the manuscript page.
Double Maths First Thing is Colin’s weekly newsletter. Usually several letters, arranged into words.
Hello! My name is Colin and I am a mathematician on a mission to spread joy and delight through the medium of mathematics. It’s Wednesday morning and it’s time for Double Maths First Thing.
Number City!
There’s only one place to start this week: on an Orcadian all-weather hockey pitch, where Katie Steckles is at it again. A crack team of maths communicators built towers of boxes representing the prime factors of the numbers up to 64. I love this sort of large-scale outreach project — low barrier to entry (anyone can doodle a number on a box), high curiosity factor (“what are all those nerds doing on the hockey pitch? Is that… Matt Parker? With SEVEN FROM NUMBERBLOCKS?!“), and plenty of depth available for those who seek it.
The only thing I don’t like is that it’s clearly not Number City — it’s a Factory Town.
A Number-Picking Puzzle
I secretly write down a number between 1 and 100 and you have to guess it. You pay me £1 every time you make a guess (I’ll tell you “higher”, “lower” or correct) and I’ll pay you £6 once you get the right answer.
How should you play if you know I’ve picked at random? How should you play if we’re both playing to win? Who wins in the long run?
The reason I ask is, Steve Ballmer (one-time CEO of Microsoft) used to ask this as a coding interview question; his answer was controversial.
There’s some analysis from John Graham-Cumming and from Possibly Wrong — I haven’t gone through it myself, because I’m still trying to figure out the answer when it’s a number from 1 to 3.
Classical maths
I’ve included the next three links as a challenge. Don’t get me wrong, I think they’re good, solid mathematical blog posts, made available for free, so I’m not going to complain — and yet I have a nagging feeling they could be done better. Could you explain these things more clearly?
Eli Bendersky has some notes on the Euler formula (I shall redact my rant about how the very idea of “the most beautiful equation” is offensive and wrong and replace it with one grumbling about Eli not properly LaTeXing up his functions).
I mean, who has? In case you’re one of today’s lucky 10,000 who don’t know about Chris Smith, he’s been running a school maths department newsletter for… well, I subscribed a decade ago and it’s gone from issue 293 to 690 in the meantime, so I imagine you can work it out. If you want to be part of the fun — a seemingly endless supply of jokes, puzzles and news, not to mention a milk rota that I’m too far in to ask about now — you can sign up by sending him an email.
In the meantime, if you have friends and/or colleagues who would enjoy Double Maths First Thing, do send them the link to sign up — they’ll be very welcome here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website.
You’re about to spend the next 25 minutes watching a guy solve a sudoku. Not only that, but it’s going to be the highlight of your day.
The highlight of my day recently was coming across Phistomephel’s ring, which is a neat consequence of standard sudoku rules.
Tony Mann pointed me at another Cracking the Cryptic video with the same energy — the frustrations and feelings of stupidity that come with not having the answer yet, followed by the sheer joy of having worked out something clever.
Back to taking pleasure in maths, here’s a short interview with Talithia Williams, PhD: I loved the bit about maths appreciation, and trying to change the mindset that maths is about doing calculations to pass a test.
Another article that caught my eye this week was about climbing. Or rather, spotting an error on the climbing wall and getting it fixed. It’s interesting for several reasons, but what grabbed my attention was what I think of as x-ray vision: the power to see that something looks off, and the insistence that it be put right. That strikes me as a very mathematical thing. (And, speaking for myself, possibly an autistic thing. Drives me MAD when people don’t care about breaking the rules, I tell you.)
Thanks to September ending on a Monday, the monthly MathsJam meet-up is coming around distressingly quickly — those that meet on the traditional penultimate Tuesday will do so on September 17th. You can find your local MathsJam here — I’ll be at the Weymouth one.
Also, if you’re planning to go to Big MathsJam in November, early-bird pricing ends on Sunday.
There’s a Finite Group livestream on Friday, September 13th at 9pm BST — Katie and Ayliean are putting the ‘fun’ into ‘fundamental theorems’, it says here.
That’s all for this week! If there’s something I should know about, you can find me on Mathstodon as @icecolbeveridge, or at my personal website.
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of August 2024, is now online at Maths for Life.
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.
Hello! My name is Colin and I am a mathematician. Welcome to issue 0 of Double Maths First Thing, in which I highlight some of the mathematical things that have caught my eye this week.
Let’s talk about \( \pi \) and powers
First up, a nod to physicists Arnab Priya Saha and Aninda Sinha for doing something with no real application: they “accidentally discovered a new formula for pi”. There’s a bit about it in Scientific American, a Numberphile video, and a paper in Physical Review Letters (open access). I’ve not worked through it in detail, but it’s got a Pochhammer symbol in it, so it must be good.
I promise this isn’t always going to be about pi, but I also stumbled on a proof that pi is irrational — again, I’ve not worked through the details, but it looks like it would be accessible to a good A-level class with a bit of hand-holding.
Via reddit, a surprisingly tricky problem with a lovely twist in the tail: show that \( 3^k + 5^k = n^3 \) has no solutions for \( k > 1 \). (There’s a hint and a spoiler over on mathstodon.)
Somewhere to visit: W5, Belfast
I’ve recently been on holiday in Northern Ireland. We visited W5 in Belfast, which is a pretty cool science museum — lots of hands-on stuff, including a build-your-own Scalextric-style car, bottle rockets and a green-screen bit where you can present the news about the alien invasion. On the minus side… there are lots of missed opportunities for highlighting the maths that underpins it all. Still, it’s a fun half-day if you’re all Titanic-ed out.
Maths in the news
In the proper news, the Guardian had a long read about Field’s Medallist Alexander Grothendieck; although it too is a bit maths-light, it’s understandable given quite how heavy Grothendieck’s maths is. Katie Steckles also pointed me at the devastating news that UK railcard discounts are dropping from 34% to 33.4%, which strikes me as the sort of thing that probably costs more to implement than it could possibly save the train operators.
![Screenshot of a PDF of the transcription. Text reads: "A watche at cardes to tell a man at what ower he thincketh to Risse proved trewe"; then a horizontal line; then "tacke xij cardes [w^th (and)] the knaue of clubes & laye them Round licke a clocke turnynge them all ther faces downward but the knave of clubes & laye hime vnder neth licke you^r watche & laye them this- then aske the ptie at what ower he will Risse & leat him kepe yt to hime seallfe & yf he thincke vij then poynte hime a a card to teall frome & bead hime yf he thincke vij to teall the card viij observinge this Rewle frome your leafte hand a cownt; that j for xv always & when you will have the ptie ^tell a poynt hime to tell toward you^r Right hand to what card you will a poynte hime you muste tell to your sealfe as before toward you leafte hand be ginynge at the j wch is xv & so tell them vpward as thus 15 16 17 18 19 20 21 22 23 24 & w^ch of al thes nvmbers that you poynte vpon bead him tell frome that w^ch he thinketh in his mind tel he haue towld to so maney toward his Right hand & then leat hime turn yt vp & yt shalbe that w^ch he thincketh a proved." To the right is a diagram of the numbers 1-12 arranged in a circle, with an X in place of 10.](https://aperiodical.com/wp-content/uploads/2024/09/image-2.png)
