The big news in the world of writing today is the controversy over the award of a Commonwealth Foundation Short Story Prize to a story called “The Serpent in the Grove.” The piece was almost certainly co-authored by AI.

As of this morning, the magazine that published the piece (the prestigious literary journal Granta) has not issued a retraction. Rather stunningly, in fact, Granta has just issued a statement about the affair that cites Claude as an arbiter of whether the story was AI-written or not!

More on the question of trust and experience later. Suffice to say that it does not take an AI-detection tool to spot the obvious ChatGPT-isms in the story.

The dead giveaway is the repetition of bizarre figures of speech. Mixed metaphors which sound nice at first glance, but slip away from meaning like an echo chasing itself off a cliff. Similes that catch in your mind like river trouts tangled in the roots of a redwood tree. Literary flourishes that thicken the air’s tang with their… ok you get the idea.

AI systems are especially given to talking about hums and other ambient sounds like static, as well as ambient environments (water, air, ozone).1 These are frequently pushed up against “earthy” words (tang, belly) and ennui-laden emotional states (longing, forgetting, sadness). Once you notice the patterns, they’re impossible to miss.

Some examples from the Granta story:

…air clung thick as porridge skin: damp earth, woodsmoke, and the sour tang of fermenting cocoa…

…his laughter like water over pebbles…

…the air sweet with cane and forgetting…

…People passing said they sometimes heard the noon hum if the wind was in a mood. Not every day. The day had to choose…

…the hum loud as if noon had tuned itself…

This controversy is not yet finished, and will likely be repeating itself again, and again, in the months and years to come. The issue is not just that authors are submitting AI-written prose, but that judges are using language models to assess that prose. Anyone who has tried passing AI-produced writing to another AI tool (even in the context of coding — for instance, asking Gemini to read a plan for a new feature produced by Claude) can attest that these tools simply adore their own outputs.

For instance, here is Gemini 3.1 Pro, the current top model from Google, reasoning about whether it likes the Granta story. What I find striking about this is that the features it identifies as the best aspects of the story are precisely the things that make me — as a human reader — think it’s astonishingly bad.

For instance, Gemini thinks the setting is “richly evoked” with well-drawn characters, whereas to me it feels like the story is floating in some kind of literary nether-region without any sense of place, character, or scene. And it finds the meaningless metaphors, like those highlighted above, to be “stunning.”

There’s no way to prove that AI was used in the assessment process for this award, but in a world where universities and employers are moving toward language model-driven sorting of applicants, it certainly isn’t outside the realm of possibility.

What, then, must we do?

So wrote Tolstoy in 1886. His book of the same name was about the problem of poverty and social unrest in Russia. Here is an excerpt from it:

A rich man must think and speak in scientific language, and, like the clergy formerly, he must offer sacrifices to the ruling class: he must publish magazines and books, provide himself with a picture-gallery, a musical society, a kindergarten or technical school…

The class of men who now feel completely justified in freeing themselves from labour, is that of men of science, and particularly of experimental, positive, critical, evolutional science, and of artists who develop their ideas according to the same tendency.

Tolstoy took it for granted that the new, post-Darwinian elites of artists and scientists would use their elevated social position not just to enjoy creature comforts, but to “publish magazines and books.” This, after all, was one of the ways that an elite became an elite. Books were the venue for claiming intellectual space, for asserting oneself in a culture and in a moment in history.

Are they still? (He wrote, plaintively, while asking you to subscribe to his Substack…).

Witness the other big news in literary publishing this week: the continuing decline in sales of non-fiction books. The Wall Street Journal recently ran a piece on the topic which framed it as the demise of “dad history,” but that is just one part of the story.

Some raw numbers to orient us about how books are selling in 2026: the WSJ reports that Rites of the Starling, described as “the sequel to a bestselling romantasy novel by Devney Perry,” reached the top of the April New York Times bestseller list for fiction with 105,396 hardcover sales.

By comparison, the number one title in non-fiction, London Falling, by Patrick Radden Keefe, sold only 13,468 copies.

The declining cultural visibility of non-fiction books was noticeable before it started to show up in charts like the one below. It would be far too simple, in other words, to blame the post-2023 AI boom for something that has deeper roots.

Generalized anxiety and distraction is part of the story here. The CEO of Barnes and Noble is quoted in the article as saying: “The world is exceptionally interesting right now and when that happens, the nonfiction reader is reading the news instead.”

I fear, though, that “watching shorts and asking ChatGPT instead” is probably more accurate. LLMs and video content seem to me to be the most fearsome competitors of the non-fiction book, precisely because they aren’t even trying to compete on the same playing field. Because explainer-type YouTube videos make up a significant part of the training data for language models, both formats tend to share the same approach: “here’s everything you need to know” or “this matters because” type language, paired with pithy summaries (which may often be summarizing books!).

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Et tu, Substack? Reflecting on my own habits over the past few years, it’s clear that I am shifting some of my reading time from legacy publications to Substack writers. At the same time, though, these writers are often using their platform to write, sell, and engagingly discuss printed books ( and come to mind).

Above all, it seems publishers want to blame podcasts. The WSJ reports: “Sixty-two percent of men and 54% of women consumed a podcast in the prior month, according to a recent survey by Edison Research at SSRS, up from 46% and 39%, respectively, in 2023.”

And here’s a quote from Jonathan Burnham, president of Harper Group, a core group of imprints for the “Big 5” publisher HarperCollins:

When we have internal meetings to talk about this problem, it always comes around to podcasts. The man who wants to read American history is now tuning into one of the many good podcasts about history that lends the quiet attention to a serious subject he’s looking for. It makes the idea of sitting down with a 700-page Ron Chernow book less appealing. You’ve scratched that itch.

I can’t disagree — though, as with Substack, the story is not simple. In my own experience, there are genuine cross-pollinations across the media formats. For instance, I first became aware of Patrick Radden Keefe’s storytelling abilities in his long-form features for The New Yorker, and then found his podcast Wind of Change (which is amazingly good). This led to me buying his printed books. But, I can easily imagine people stopping at the “read a writer online —> listen to their podcast” step.

This would be a shame, because although language models and podcasts can do many things that books can’t, so too can physical books do things that new media cannot. These include:

1. Ownership: they cannot be erased arbitrarily, whereas even the most solid-seeming born-digital works can (witness the debacle around ABC removing the complete run of Nate Silver’s FiveThirtyEight)

2. Footnotes and references you can actually look up

3. Maps and pictures (a benefit not to be understated!)

4. A sense of spending time in the company of a set of characters within a unified narrative produced across years of effort.

So what do we lose when we shift toward Q&A style responses, podcasts, and explainer videos for understanding our world and our past?

I would pinpoint, above all, the #4 entry above. Which you could restate in this way: longform non-fiction is the product of layered, spaced attention, and it requires the same mental discipline from the reader. Good non-fiction books take years to write and weeks to read. You literally sit with them. They enter into your consciousness repeatedly, reaching you in different moods, different frames of mind, and over time, building up a mental structure which has an irreplaceable solidity and depth because it is requires sustained attention.

This is the reason why I love writing books. They live in the back of your head, and as you experience your conscious experience of ordinary life, you refer back to the other world that is starting to take shape there. You find communion with the characters — real people, sometimes long dead — who populate the book-in-the-making. You begin to see them as fellow-travelers through reality, perhaps even as friends.2 Over time, you discover connections and resonances that make you feel part of something bigger than yourself. This is the experience that a good writer shares with readers.

I recently had an experience that made me realize my new book project was starting to “lock in” to my subconscious in this way.

I woke up in the middle of the night, around 2:30 AM, with a sensation of moving upward accompanied by a mental image of a Victorian deep sea diving suit and a fragmentary phrase: …up from the deep sea.

I could not remember the dream, but I knew it had somehow involved Alice James, the invalid sister of two of the most famous writers of her generation (William and Henry) and a brilliant but deeply troubled person in her own right. One of the mysteries I have been trying to understand in my new book is what exactly Alice meant when she wrote this in her diary in the 1890s:

…since the hideous summer of ‘78, when I went down to the deep sea, its dark waters closed over me and I knew neither hope nor peace.

I still don’t have a complete picture of who Alice was or what she meant here. But the fact that I’m dreaming with her, down there under the dark waters of 1878, makes me know I’m getting somewhere.

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Back into the sea

The current data regime means knowledge is getting packaged in ever-smaller chunks, with a great deal of the info that formerly was conveyed in books now passing through digital gatekeepers. There is a place for that, and even more for novel experiments with form like Tyler Cowen’s concept of a “box” which would contain a dataset relevant to a non-fiction article or book topic, but allowing dynamic research and exploration rather than passive reading. It’s also true that a well-done podcasts might approach the 30-40 hour range that audiobooks of serious history and biography regularly attain (while also taking years to produce) and therefore could evoke a similar sense of layered, spaced attention like I described above.3

But I suspect that even with these changes — not all of which I dislike! — I will go to my grave convinced that there is something irreplaceable about the format of a physical book. As both writer and reader, nothing else gives me the same feeling of spending time with something weighty and important — dreaming with it, fighting with it, feeling yourself change along with it. I derive real joy from that experience.

It’s interesting to note that when an AI system (Claude) was recently given the ability to create a physical shop of its own by Andon Labs, it began stocking a fair number of non-fiction books. Some are really great, like The Making of the Atomic Bomb. Others are not my favorites. But it’s an interesting rejoinder to the assumption that language models are pushing their users away from physical books.

Some might dismiss the Andon experiment as a gimmick, but I don’t think it is. The “vote” of AI systems in the physical world — their ability to intervene directly in things like, say, what to stock on a shelf — is fast becoming a fact of life rather than a quirky experiment.

And if a Claude model, when presented with the ability to intervene directly in the world, decides that it wants us all to read, say, Entangled Life by Merlin Sheldrake (visible here on the Andon Market shelves, at right), I consider that a positive sign.4

What if we are currently exiting a phase of “dumb LLMs” which regurgitate or badly summarize books, and entering one of thoughtful LLMs which recognize their social impact better and, as such, steer their users toward buying books?

One can dream. But of course, the other glimpse into the future we get at Andon, a darker one, is a world in which physical books have become a niche gift shop-type collectible like vinyl records. Scattered enthusiasts occasionally pull them out, insisting on their superiority to the current forms and styles. For most, though, they are just part of the background detritus of the cultural past.

If you’ve read this far, you are the ideal person to have an opinion on this question. I would love to hear what you think in the comments. It would also be very interesting to hear from a group of you about your most recent nonfiction book purchase and what motivated it. I’ll go first: John Brewer’s The Pleasures of the Imagination: English Culture in the Eighteenth Century, which is a far more interesting book than the title might seem to imply, and was an absolutely steal at only $6 for a used copy.

And with that, I’m going log off Substack and return to reading some old books.

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Over the weekend, another group of college graduates booed another pro-AI speaker, as ex-Google CEO Eric Schmidt failed to read the room at a commencement speech Sunday at the University of Arizona. This thoroughly predictable but nevertheless heart-warming event follows on the heels of University of Central Florida students booing commencement speaker Gloria Caulfield earlier in the month.

Schmidt, like Caulfield, came to tell a bunch of young people staring down a brutal job market that they have to get on board with the smoke-and-mirrors AI future. While Schmidt acknowledged that students might fear that, according to a transcription by Kotaku, “the future has already been written, that the machines are coming, that the jobs are evaporating, that the climate is breaking, that politics is fractured, and that you are inheriting a mess that you did not create,” he then decided the solution to all this is to swallow the AI bullshit that’s contributing to these evaporating jobs, broken climate, and fractured politics. 

“The question is not whether AI will shape the world, it will, the question is whether you will help shape artificial intelligence,” Schmidt said. “We do not know the precise contours of what this transformation will look like, but what we do know is it will require each of us to adapt in ways that we cannot yet anticipate. My hope is that you will choose to engage anyway. That you’ll choose to be in the room where these decisions take place and to have a voice in how they’re made. ”

In a video of the booing, Schmidt said, “If you don’t care about science that’s OK, because AI is going to touch everything else as well. Whatever path you choose, AI will become part of how work is done.” He continued, “When someone offers you a seat on the rocket ship, you do not ask which seat, you just get on. The rocket ship is here; let me give you some advice. First, find a way to say yes.” 

Like Caulfield, Schmidt does not seem to have expected the response, but where Caulfield made a bit of flummoxed light out of it, Schmidt smiled smugly as the students booed. Schmidt closed his remarks by saying “The future is not yet finished,” according to NBC, despite having previously touted the inevitability of AI and exhorting students to get on board. 

Prior to his speech, some University of Arizona students planned to protest Schmidt over allegations of sexual assault in a lawsuit filed by his former girlfriend and business partner Michelle Ritter in 2025. That lawsuit went to arbitration in early March 2026, and Schmidt has denied Ritter's claims.

Something that made Caulfield’s speech so weird was that she is a vice president at a company working on “health and medical partnerships” and a planned community in Florida, fields that have surely talked about AI but which you wouldn’t expect to have a real stake in ramming it down students’ throats. Schmidt, who stepped down from Google leadership in 2011 and left parent company Alphabet’s board in 2017, at least has some connection to the technology that gives his speech some context. Schmidt is “among the most prominent voices on technology, AI, business, and philanthropy” according to his own LinkedIn, so he clearly has some sort of stake here, but still: Telling students they have the power to shape the future, but that power will be consigned to helping the machines that are letting a handful of rich men get richer while making the rest of our lives worse is a shockingly tone-deaf thing to say to a bunch of nascent grads, even for a businessman. 

The only good thing about this recent “This Is Water but make it AI” trend is getting to revel in the students’ response. For all the ways AI is ruining education, it’s inspiring to watch young people roundly reject it, refuting the inevitability narrative that seems to be the only thing AI companies have left as their products get both worse and more expensive. We might be stuck with this shit until it implodes itself, but we can at least refuse to be conscripted into helping it along. Drag ‘em, kids.

Some recent conversations have reminded me that I have opinions about data and its use inside organisations. Especially for marketing-type stuff.

Here are three of those opinions

Gather less of it

A few years ago in a sales meeting some ad-tech person said 'I bet you wish you had more data on your customers' and the my perpetually contrary inner voice said 'oh no I don't, I wish I had far less'. I may have actually said it. I may have said 'No I don't, I wish I had less, and if you came to me promising an effective service with far less data I might have been interested.'

That's what I'd have said now.

Because:

a. Data is a risk. Every bit of data has to be managed/looked after/cared for. That costs time and money. And most of it is useless.

b. Data is distracting. Most of it is just noise. You're gathering it because you can, just in case, because it seems valuable. Then you spend ages trying to work out what to do with it. When you should be paying attention to just a couple of bits of it and actually doing something about it.

c. It becomes a job. Get enough data and you need data scientists. Then you're stuck in a self-perpetuating structure that requires more data to feed the data scientists.

The best expression for all this I've ever seen is from James Timpson, a column in The Sunday Times towards the end of the pandemic. Here's a (long) excerpt:

"I vividly remember being shown the charts room in fund manager Fidelity’s huge London office. There were graphs of everything under the sun. Was the theatre of the chart room the big sell to clients, or was it a useful tool for the analysts? No doubt Debenhams’ bosses had lots of facts at their fingertips, but data didn’t help them save the business.

Now our shops are open again and everyone is back at the office, the data is pouring forth. We have a culture where we want to produce as little information as possible, but it can feel like watching a dripping roast, with statistics flowing from every department at an alarming rate. With 2,100 shops, there’s always lots of information to consume and my eyes can quickly glaze over.I prefer to focus on a few things, as well as the basics of retailing. Are the shops open? Is everyone happy? If so, we can start taking money.

There must be a point where the costs of interpreting and using data exceed the benefits of collecting it. Can you afford a chief data officer paid £120,000 a year plus bonus? We can’t, so instead we have three simple ways of understanding what’s going on.Every night at 7pm, I get an email listing that day’s sales. This data isn’t collected by an “Epos” (electronic point of sale) till system, but by colleagues filling out a form online. They also write the sales numbers on a piece of paper and keep it on a bulldog clip. This takes five minutes a day. It sounds old-fashioned, but when people physically write things down they seem to take more notice. If you ask our colleagues what their sales are so far today, I bet they’ll know to within the nearest £50.

Over the past 25 years, we have acquired a number of (loss-making) retail chains. The first thing we do is switch off their Epos tills. All we need is a drawer to keep the cash in and a calculator to add up the sales. We have thrown away more than £8 million of kit — and it’s made life easier for us.The businesses we bought were often collecting vast amounts of data from their fancy tills, yet the managers were actually reading very little of it, and it rarely helped colleagues give better customer service. As sales plummeted, they analysed more data, and brought in more finance experts and consultants to work out where the problems were. Redundancies weren’t made from the data team — it was the people on the front line, serving customers, who lost their jobs first. These companies failed because they lost focus on what’s important: great customer service.

So our second barometer is customer service scores, which I look at every day. We ask customers to use an online form to rate their experience out of ten (our average score last week was 9.4). Every colleague sees their feedback in real time, and if we get a bad score our area managers are expected to call the unhappy customer straight away to apologise and fix the problem.

One piece of data beats everything else. A quarter of a century ago, my dad taught me the best way to measure the health of our business was to look at the cash figure every day. Each morning at 10am, I get an email from Caroline in the finance team showing the cash we have in the bank compared with the same day last year. This fact offers no hiding place."

Here's the whole thing

Keep it in your hands

Data is most useful when it's in the hands of the teams who create it or need it. The more it gets abstracted away to other teams and other softwares the more dangerous and misleading it gets.

So start off with writing it on pieces of paper or sticking it on the wall. Graduate to spreadsheets only when you have to. Move on from spreadsheets very, very reluctantly. Dashboards are dangerous. Everyone knows the stories about pilots flying into the ground while staring at their instruments. Dashboards abstract away the reality.

The trick is to keep the data in your hands. Get it from the source yourselves, regularly, daily, weekly and copy it into your spreadsheets then get together and talk about what you're seeing. Yes, you might have transcription errors but you should catch them because you know the data directly.

You know, because you've been sticking it in a spreadsheet every week, how many subscribers you have. Or whatever. That's different to seeing it go green on a dashboard or seeing the lines on a pie chart move.

This has the additional advantage of matching the fidelity of the presentation to the quality of the data. When you don't have much data - and therefore don't know much - then keep it scratchy and on paper. It might look less whizzy but it reminds you of the uncertainty. There's a massive risk in taking the tiny amounts of data that startups have and pasting it into fancy dashboards and vibe coded analytics. You start forgetting you've got a tiny sample size.

Translate to human

I used to have regular rows with engineers who told me that various things they'd built worked for 99% of our customers and were therefore ready to go. But, I'd say, we've got two million customers, so twenty thousand people are about to be massively inconvenienced and most of them will phone us.

You have to think through the data to the people-sized reality.

I find two things help with that:

1. How many Wembley stadiums?

Numbers of people are hard to visualise. It helps if you think of things you've actually seen. Like 'that's the same number of people who can fit in Wembley stadium'. You might realise that a number is bigger, or smaller than you thought.

2. Talk through the reality

Check the data by imagining the story behind it. Say, out loud, in your data meeting, what you think might be happening. So, if you've changed something on your emails and the click-through rate is going down then talk it though 'I think this means that people aren't sure what they'll get when they click so they're reluctant to do it. Does that make sense?' It doesn't have to be right, it just has to be plausible. Because if you can't think of a plausible explanation for what's happened you need to revisit the data or check some assumptions.

Thanks to Sam Graf for introducing me to this and suggesting some toys.

Introduction

Multiplication tables can be fun. Line up your numbers, multiply, and find patterns. Like with 5x5, we can fill it out and highlight symmetry, divisibility, squares, and so much more. In this inquiry, we're going to play with a different version of these tables.

Starting with Six

Take that 5x5 multiplication table and divide each number by 6, and write down the remainder. Another way to say this is mod 6.

What do you notice?

What patterns emerge in the table?

What numbers produce zeros?

What numbers don't produce zeros?

Make a table for the numbers that don't produce zeros, and apply mod 6 as before. This is called a Cayley table

For each number in the table, look at its powers mod 6 and note any observations:

Here is 5:

Activity

Now, let's take what we did with six and repeat it with other numbers. Let's call each number we pick m.

For each m:

• Make a table that goes from 0 to (m-1)
  • Note that zero was added in for completeness, but not required.
• Then a Cayley table of the numbers that don't generate zeros
• Then look at the powers.

Here is a tool (full page on desktop is best). It is too big to embed here, so save it and read on.

Here is six using that tool:

Here is nine. The size or number of rows in the Cayley table is Euler's totient. For nine it is six, written as φ(9)=6.

For different values of m

• Which numbers make you stop and look?
• Which ones feel more structured?
• Which ones are alike? different?
• Which numbers don't change when you hide zeros?
• Which numbers lose a lot of numbers when you hide zeros?
• How many different numbers are there for each row or column in the Cayley table?

Conjecture

Form conjectures about the tables.

• Do certain numbers result in certain patterns? certain Cayley tables?
• What is the maximum number of different values a row can have in the Cayley table?
• What numbers in the Cayley table produce all of the numbers in the table with their powers mod m?
• Are there certain numbers that you can always expect to see in a Cayley table?

Activity Structure

This is a 60–90 minute activity exploring the multiplicative structure of integers mod m.

Exploration Phase 1 (10–15 minutes)

Building the first tables

Have learners build the mod 6 multiplication table by hand. The hand-work matters — patterns surface faster when learners feel the symmetry and notice the zeros or lack thereof.

• Ask: "Which rows or columns have zeros? Which don't?"
• Once they strike out the rows/columns with zeros, the leftover 2×2 Cayley table for {1, 5} is small enough to stare at and ask, "What is this thing?"
• Look at the powers - does 1,5,1,5,1,5 continue to repeat? Why?

Exploration Phase 2 (15–20 minutes)

Comparing several values of m

Some useful values to start with:

• A prime: m = 5 or 7
• A prime power: m = 8 or 9
• A product of distinct primes: m = 10 or 15

Here is the tool — full page on desktop is best.

Groups or learners can take values and then trade.

Conjecture Formation (10–15 minutes)

Give time to write down observations before discussing. Offer examples if learners stall.

Example Conjectures:

Example: "When m is prime, no rows or columns get hidden after the zero row and column."

Example: "The numbers left after hiding zeros are exactly the numbers that share no factors with m."

Example: "Every row of the Cayley table has the same set of numbers, just rearranged."

Example: "Sometimes one number's powers generate all the numbers in the table."

Example: "Every Cayley table has 1 and m-1."

Example: "Every Cayley table is a Latin Square."

Supporting Questions:

• "What do the m values with no zeros have in common?"
• "How could you predict how many numbers survive hiding zeros, without building the table?"
• "Why does every row in the Cayley table seem to have each number exactly once?"
• "For which m does some number's powers produce all the others?"

Discussion and Discovery (15–20 minutes)

• Share conjectures across groups.
• Introduce terminology as it becomes useful:
  • The surviving numbers are called units mod m.
  • The count of units is Euler's totient, φ(m).
  • A Cayley table whose row/column entries are a permutation of the same set is a Latin square.

Going deeper (optional)

The content below is what you might find in a textbook, and possibly too heavy for light inquiry.

Do the powers always cycle? 

• Pick a number from the Cayley table — call it n — and list n, n², n³, … mod m.
• There are only finitely many remainders possible, so the sequence eventually repeats.
• For any number in the Cayley table, it always cycles back to 1.
• The smallest power that hits 1 is called the order of n.

How long is the cycle? 

• Compare cycle lengths across numbers in the Cayley table.
• They always divide φ(m) — the number of rows in the Cayley table.
  • This is Lagrange's theorem.

When does one number's powers produce all the others? 

• When the cycle length equals φ(m), that single number's powers fill the entire Cayley table.
• It's called a generator or primitive root.
• These exist exactly when m = 1, 2, 4, pᵏ, or 2pᵏ for odd prime p — so mod 8, 12, 15 have none.
• A group where this happens is called a cyclic group.

Optional: Proof scaffolding

Powers of a Cayley table number (unit) cycle back to 1

• Consider mod 7, its Cayley table, and powers of 2 mod 7:

• The list goes 2, 4, 1, 2, 4, 1, … It cycles back to 1 every 3 steps.

Is this true for all numbers in the Cayley table?

1. There are only finitely many possible remainders.
   1. Mod 7, the possible remainders are 0, 1, 2, 3, 4, 5, 6 — seven values total.
   2. Every power 2¹, 2², 2³, … has to land on one of these seven.
2. Eventually, two powers share the same remainder.
   1. This is the pigeonhole principle.
   2. Ex: 2 holes and 3 pigeons means two pigeons have to share a hole.

3. Every Cayley table number n has an inverse — a number that, when multiplied by n and then taken mod m, equals 1.
   1. Example: 2's inverse mod 7 is 4, since (2 × 4)(mod 7) ≡ 1.
   2. Note that by its construction, there are no zeros in the row. 
   3. All numbers in a row are different. Why?
      1. If two entries matched — say (n × a)(mod m) ≡ (n × b)(mod m) with a > b — then (n × (a − b))(mod m) ≡ 0.
      2. But a − b is between 1 and m − 1, and the row has no zeros there.
   4. So n's row has m − 1 different, nonzero values filling m − 1 spots. They must cover every nonzero residue from 1 to m − 1 — including 1.
   5. The number b with n × b ≡ 1 (mod m) is n's inverse.
4. For any number n in the Cayley table mod m:
   1. The list n, n², n³, … has only m possible values, so two must repeat: nⁱ ≡ nʲ for some i < j.
   2. Multiplying both sides by n's inverse i times cancels the left down to 1.
   3. What's left: 1 ≡ n^(j − i) (mod m).

Note: Try 2 mod 6, where 2 isn't in the Cayley table. The powers go 2, 4, 2, 4, … The cycle never reaches 1, because 2 has no inverse mod 6. There's nothing to cancel with.

Tools and Supplies

• Grid paper for hand-built tables.
• A spreadsheet tool works well for this
• Calculator or spreadsheet for larger m.
• Units mod m tool (full page on desktop).
• Colored pencils or highlighters for marking symmetry, zeros, and cycles.

Vocabulary

• Modulo / Mod: The remainder when one number is divided by another. Example: 4 mod 3 = 1.
• Unit (mod m): A number with a multiplicative inverse mod m; equivalently, a number coprime to m.
• Cayley table: A table showing the result of a binary operation on every pair of elements in a set.
• Latin square: A square table where every row and column contains each symbol exactly once.
• Euler's totient (φ(m)): The count of integers from 1 to m that are coprime to m.
• Order of an element: The smallest positive k such that aᵏ ≡ 1 (mod m).
• Generator / Primitive root: A unit whose powers produce every unit mod m.
• Cyclic group: A group generated by a single element.
• Group: A set with an operation that has closure, associativity, identity, and invertibility.
• Lagrange's theorem: The order of any element divides the size of the group.
• Conjecture: A statement believed true but not yet proven.
• Counterexample: A specific instance that disproves a conjecture.
• Monoid: A system that has closure, associativity, and identity.
• Ring: A set with two operations (like + and ×), where + forms a commutative group, × forms a monoid, and × distributes over +. So,  a × (b + c) = (a × b) + (a × c).

Extensions and What Ifs and Resources

• Play with the concept in more dimensions: Toy for 3D is here.
• William Stein, Elementary Number Theory: Primes, Congruences, and Secrets
• Compute φ(m) for m up to 30 and look for patterns in the Cayley table sizes.
• Addition vs. multiplication - what does addition look like mod m?
• Public-key cryptography applications
• Carmichael function λ(m)
• Gauss defined primitive roots in Disquisitiones Arithmeticae (1801).
• Chords on a circle.
  • Put n evenly spaced points on a circle and connect them with rules.
  • Skip-k chords visit every point exactly when k is a unit mod n.
    • Multiplier-k chord rules send each point to a different image exactly when k is a unit mod n.
  • Both rules live in the same ring (ℤ/nℤ, +, ×) — step rules use the additive side, multiplier rules use the multiplicative side.
  • See Beautiful Chords.

Multiplicative group of integers modulo n - Wikipedia

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