There’s plenty of legitimate complaints about academia. I’ve made them myself. But is it too much to hope for that one day, just maybe, people will stop falling for the availability bias?

is all fired up about Christopher P. Barton. So let’s look into Mr. Barton: PhD Temple, author of Trowels in the Trenches (University of Florida Press), has no current academic affiliation that either I or Claude could find. He seems to have once been employed as an assistant professor at Francis Marion University in South Carolina, but is no longer on their website. Although his LinkedIn page says he an adjunct instructor at Widener University, the Widener website does not list him and I don’t know if LinkedIn is current. As far as I can tell, Barton is either not in the academy at all or just clinging on. His Google Scholar page says that in 14 years he has been cited all of 154 times. Absolutely no part of his story is “elite.” Not to be too snobby about it, but Barton barely reached the bottom rung of academia and seems to have slipped off of that.

Nevertheless, Barton’s LGBTQ+ cavemen take must surely be indicative of the whole woke activist cult field of archeology, right? Well, Antiquity is archeology‘s flagship journal. Here are all of the articles in the latest issue:

• Understanding a first attempt at pottery: early fired containers at Çemka Höyük

• Human maritime adaptation on Haitan Island, China, c. 7500 cal BP

• A fortified Early Bronze Age settlement in the Southern Caucasus

• The Pocó-Saladoid connection and the first widespread ceramic tradition in Tropical Lowland South America

• Uncovering the lives of rock doves (Columba livia) in Late Bronze Age Hala Sultan Tekke, Cyprus

• Reconnecting the dead in Iron Age Britain: funerary processing at Loch Borralie, Scotland

• Failure or resilience? Community investments at ancient Roman Veii

• Genetic relatedness, social status and cemetery organisation: the Xiongnu Tamir necropolis, Mongolia

• The death jar: a new mortuary tradition at the Plain of Jars, Lao PDR

• Radiocarbon dating of textiles preserved in late medieval helmets from Benicarló, Spain

• Surgical anaesthesia in Ming China: aconitine residues on medical instruments

• Mediterranean maritime visibility: old limits and new approaches

Hmmmm, I’m not sure which one of these is the absolute, cultish, woke activist-style dross that academia is apparently awash in. Probably all. But hey, that’s not my field. What’s happening with us philosophers? There’s only two articles (plus a bunch of book reviews) in the latest Philosophical Review, arguably our leading journal: “Could the truths of mathematics have been different?” and “Advancing the Boltzmann Brain skeptical challenge.” I’ve not read them, but based on the abstracts they don’t seem too political.

Wait, I thought this stuff was everywhere. No? Klinger is certainly looking for it:

Let’s apply this methodology elsewhere. Pick a random company. How about Comcast/Xfinity? Is Comcast/Xfinity a terrible company? Send me all your horror stories about Comcast/Xfinity! Ok, a Google search of “Comcast is terrible” yielded some excellent results. Try it yourself. There’s whole Reddit groups devoted to why they are the worst, a Wikipedia page called “Criticisms of Comcast,” a Facebook group called “Comcast Customer Service Sucks,” and the list goes on.

Maybe they are particularly bad and I’m just cherry picking. Let’s look at, I dunno, McDonalds. I searched “McDonalds is terrible” and found a Reddit group called “When did McDonalds get so bad?”, articles on the downfall of McDonalds, how their food quality is declining, lists of things customers hate about McDonalds in 2026, and plenty more. There’s loads of complaints about McDonalds.

Wait, here’s an idea. Klinger’s Substack bio says he is a lawyer. Everyone loves lawyers, right? Let’s see… searching… Huh. “Lawyers contribute least to society, says US poll.” Who woulda thunk? That’s the US, though. What’s the UK take on lawyers? The IPSOS Veracity Index is a long-running UK survey on trust. Check out these results.

Lawyers are trusted much less than professors, even less than a random person on the street. Barely half of UK citizens would generally trust lawyers to tell the truth. As Steven Wright joked, 99% of lawyers give the rest a bad name.

My point is that it’s a big world out there, and if you go looking for something, you’re going to find it. You can probably find people who hate Dolly Parton. The very reason that researchers (eventually) came up with double-blind testing was to make sure that they didn’t merely see the results they were wanting to see. Previous scholars weren’t dishonest; they just didn’t know that their subconscious minds were putting a thumb on the scale of evidence.

A modern way social scientists inadvertently fool themselves is p-hacking. That’s when they root around in the data until they see something that looks like a suitably improbable correlation and presto, publication. After a while people realized this was just another way of finding spurious results when they were hoping for meaningful ones. The fix—hypothesis preregistration—is like double-blinding: a way to keep everyone honest and avoid cherry-picking chance correlations.

Is there dumb wokeness in the academy? Of course. But if we are truly interested in whether it is prominent, representative, isolated, confined to only some disciplines, ubiquitous, more management than faculty (or vice versa), student-driven, or we’d like to know its etiology and causal effects, we are going to need a more sophisticated methodology than repeated applications of the availability bias. Academia is not a monoculture, as much as those on the outside try to fit us into one.

I get that Klinger is mad that his alma mater ever hired Jason Arday, and it is understandable that people are still steamed about him, but those who are claiming Arday was lynched (e.g. here and here) are embarrassing themselves. In fact, I think an accusation of lynching is the real racism; it denies human agency to Arday, portraying him as a mere passive victim of others. In reality he was a successful con man who quit his job and (apparently) took his own life when his house of cards collapsed. Those are actions he took.

Con men succeed because they are good at what they do. They identify people’s blindspots and biases and exploit them. If we really want to improve the epistemic commons we have to employ better methods than continuing to fall for known biases and inferential fallacies. C’mon, can’t we ever grow out of being toddlers who think every emotion-generating story represents all of reality? If there is one thing I have learned as a philosopher, it is this: knowledge is hard. So let us do the hard work of lashing ourselves to the mast and sailing past the sirens as they sing the call of the availability bias.

I’m teaching MPTC again this semester and have been developing a few workflow slides. Here’s a popular one.

Now if you’ll excuse me there’s a mandatory webinar I need to click through.

I imagine everyone here has received an unsolicited message telling them that their tax is overdue and that they urgently need to visit Genuine-Tax-Payment-Website.fart or that a parcel is delayed at customs and you can pay a small sum for its release at Almost-The-Right-Acronym.ak

You know it is a scam. Most people just mark as spam and move on with their day. But a significant number of people don't. They hastily visit the site, tap in their credit card details, give it their mother's maiden name, confirm address, upload a nude selfie, and only then realise that they've been had.

The Internet works at pretty close to the speed of light. You can register a .uk domain and a minute later it's accessible from the other side of the planet. Brilliant for users who want to quickly launch a website. Also brilliant for abusers who want to launch a spam campaign.

By the time enough people have reported the scammers' domain as suspicious, it is too late. In the time it takes for a registrar to disable the domain, or for its name to make its way to the Safe Browsing List, a million messages have already been sent and enough people have handed over their details.

We're told that "the purpose of a system is what it does". At the moment, the Domain Name System's purpose seems to be a vector for criminals to run scams on people at a terrifyingly high rate.

How big is this problem?

BIG!

There's a great blog post by Andrew Campling which reports on this startling claim:

The study found that at least 10% of all new gTLD domain names registered during the year had subsequently appeared on security blocklists by the time of analysis. It estimated that, taking account of subsequent blocklisting and associated domains not themselves blocklisted, the share of names registered by malicious actors may be closer to 20%.

DNS Abuse and Criminal Infrastructure: Beyond Definitions and Blocklists (emphasis added)

That links to a presentation by Interisle which contains some rather shocking statistics (albeit with disputed methodology). It looks at generic Top Level Domains (gTLD) - those are things like .com and .fun rather than country code TLDs (ccTLD) like .uk and .de.

It says 85 million new registrations of gTLDs were made in 2025. Of those 8.5 million were added to blocklists by May 2025. It reckons that a 10% abuse rate is the likely floor for these numbers and it's probably closer to 20%. One in five newly registered domains with a gTLD are scams. That's a bloody crisis.

13 TLDs had more than 50% of their registrations blocklisted.

I can understand why .bid and .loan are popular with scammers. But why .mobi?! What did I ever do to you, eh?

Who are the scammers registering these through?

Ah, our old friends at NameCheap. See Why do scammers love NameCheap?

If those five registrars had more effective policies, it might significantly dent the scammers' ability to ply their devious wares. Or they might just move on to other registrars.

As the report points out:

suspension rates for blocklisted domains were 7.4% to 16.3%.

The full report is on the Interisle website.

What can be done?

I don't know.

In the first instance, it might make sense for registrars to do strong Know Your Customer (KYC) checks on anyone buying a domain. But that stops anyone who wants to anonymously register I-Hate-Nintendo.whatever without risking the wrath of Intellectual Property lawyers.

Also, criminals have access to stolen money and stolen cards. They can convince a hapless mule to register a domain on the criminals' behalf.

Registrars could ask for an escrow payment. Pay €9 for the domain name put €900 in escrow. If your domain appears on a blocklist within the year, you forfeit the money. Criminals with stolen funds are unlikely to care but it would probably put off lots of people from getting a new domain.

There are various banned words and phrases depending on the TLD. For example, South Sudan has a list of political words which they don't want associated with their .ss ccTLD.

But if one gTLD bans a word, a different one might not. A scammer doesn't care if the gTLD is .arse or .elbow - they just want the start of the domain to look legitimate.

Some registrars have strings that they don't allow. In fairness to NameCheap, when I tried to register dwp-payments-gov-uk.pizza it told me that domain was banned. It wouldn't let me get any gTLD with that name.

But all it takes is one registrar to be slightly lax and the scammers get through. Increasing the complexity of the rules is also a hell of a burden on smaller registrars.

Besides, it's pretty easy to get a generic enough looking domain and stick the confusing bit on a subdomain. Here are a clutch mentioned in the report:

• https://gov.uk-dwpaph.bond/uk/
• https://gov.uk-dwpcjh.bond/uk/
• https://gov.uk-dwpclc.bond/uk
• https://gov.uk-dwpclw.bond/uk
• https://gov.uk-dwpclj.bond/uk/

Perhaps there ought to be a delay before a new domain goes live to allow people to object to it? That would give governments, banks, delivery companies, and a dozen more "important" organisations a right to veto any "dodgy" looking domain.

But suppose someone wants to register gov-uk-stole-my-horse.horse to protest the government's cruel policy of stealing horses - is that a legitimate use of a domain? What if the Darwin Pensioner Divas - a group of elderly singers - want to take payments for their new album of goth/punk covers, can the DPD delivery company veto dpd-payments.music?

Do we want a domain name system where powerful companies control exactly which domains we can register? If I have an idea for a domain on a Friday night do I have to wait until Monday before it can be launched? Are those companies realistically able to parse millions of domains per year and have a low false-positive rate?

All of these things are possible - but all of them come with an impact on legitimate users. To be clear, I don't know what the right answer is.

What is ICANN doing about it?

Lots! It has been a few years since I've been to an ICANN meeting, but even back then the topic of abuse was high on the agenda. They appear to be looking at ways to coordinate abuse reports between various entities, along with some other policies which should hopefully work.

There are two salient points from one of the discussions held at the recent meeting

If anybody thinks that in our current age of AI and as we move into different kinds of computing, DNS abuse is going to numerically stay steady and we will have a downward effect on that baseline 2027 number. I'm not sure that that's an accurate assumption. I think it's going to be the other thing, which is […] it's going to be easier to abuse the DNS.

And

Abusers are going to abuse because it's just too lucrative, because no matter what we do, they will find the way to make profit off of that, and will try to circumvent everything that we do. That is not a reason not to do it, though.

Quite!

As I said, I don't know the answer to this. What I do know is, much like Android's app ecosystem being a haven for scammers, DNS is facing a crisis. When trust in a system goes, only chaos follows.

I don't want to live in a world where I have to show my passport and pay thousands of pounds to register a domain which is only available after being vetted by private interests. But I also don't want to live in a world where scammers have effectively no deterrent from abusing millions of people.

The purpose of a system is what it does. I hope DNS's purpose can become less dangerous while still remaining open.

TL;DR: Metaphors like "a sinking ship" are often used to describe codebases, but are misleading. A business will sink long before code quality reaches a hypothetical floor. Technical debt has no bankruptcy, no clean reset, so metaphors that imply an end provide a false sense of security.

Software is in the domain of the abstract. It is not like a building, or a bridge, that is in the physical realm where you can see and feel the nature of the thing. If you continue to add floors and rooms to a building forever, it will collapse. Software faces no such constraint. The code can always get worse. There can always be a new layer of indirection or a reduction in performance.

1: Boarding a sinking ship

Over a decade ago I had my first encounter with an ugly legacy codebase. I had just joined Amazon as a "Software Development Engineer", fresh out of university, and worked on a team that owned code related to processing orders. On the surface, what our code had to do seemed simple. Processing an order involved writing some things to a database and calling into services owned by other teams, either to ask validity questions or to update bookkeeping on their end. My colleagues and I estimated that a sufficient implementation of this system shouldn't need more than two dozen strong engineers to maintain and evolve. Yet, our organization was hundreds of people, and the system had grown so large and complex that it had become impossible to learn how it all worked.

It was rare to stay longer than a few years in this org, and institutional knowledge had eroded. This resulted in code that was full of "haunted graveyards". Fear suppressed any (under-rewarded) efforts to simplify existing systems. The business rules for what had to be done for each type of order were decided by people long ago who weren't around anymore. These rules could sometimes be found in a hopelessly out of date file proudly calling itself a 'living document', but often the rules simply were not written anywhere we could find. Tracing behavior yourself wasn't easy either because much of the system lived across team boundaries where the code was not easy to access.

When some obscure process wasn't happening with an order that should have been happening, our pagers would angrily notify us that someone in our tangled web of service dependencies was unhappy. As a result of this feedback mechanism, the system stayed afloat, but remained difficult to change and had abysmal performance. Despite this, new layers were constantly added to support the latest Amazon products and features. This felt unsustainable, and this feeling is what makes me and others reach for a "sinking ship" as a metaphor to describe an organization that doesn't pay down their technical debt.

To their credit, there were always ongoing attempts to fix the architecture, and these usually went as follows: A new manager or senior engineer joins the organization and observes that "things are bad". Leadership at the organization agrees and wants to make things better, but there are no engineers available, so each fix attempt includes adding new engineers and teams.

The re-architectures were always a failure: The system required years of study to properly understand and was constantly changing. It's not politically viable to take so long to design a fix for the system, so naturally everyone attempting to fix things must work with incomplete information. Some of this impatience is from within: If you are trying to design a grand fix for such a prominently painful architecture, you are doing it in part because you want a promotion (and you don't want to wait too long for a promotion).

Each cycle would end with the remains of the new attempt permanently grafted onto our architecture and the leading engineer having departed with their requisite promotion. The increased headcount stays because the migration plans are too painful and unpopular to actually finish. The cycle continued as it had long before I had arrived. The sinking ship seemed to have no end.

2: Where does it end?

About three years after I had left, I was chatting on the phone with a former colleague from that team. He had recently left the company after an impressive 6-year tenure and had witnessed the cycle complete again. We were commiserating, and both of us reached for the sinking ship metaphor to describe the org, despite us having left years apart.

"Where does it end? How does it end?" he asked me, curious to hear my take on what would happen to that org in the future.

The question and metaphor didn't sit right with me, and I realized the question conflated two things. Are we talking about the code, or are we talking about the company?

A business can sink. Bad software is a real drag on the business, but how much that actually matters depends on a lot of factors. For a company with plenty of cash flow like Amazon, they can tolerate some bouts of internal rot here and there before it has any meaningful impact on their bottom line. For another company whose business model is more sensitive to software quality, bad software may be a latent invitation to a competitor to deliver the metaphorical hull breach (and no, LLMs don't change this).

For the code, the sinking doesn't end. It's an infinitely sinking ship, because there is no limit to how bad code can be. You didn't escape a building that was about to collapse. It is in a constant, neverending state of collapse. There's something wrong with using words that imply there's an end.

Software is in the domain of the abstract. It is not like a building, or a bridge, that is in the physical realm where you can see and feel the nature of the thing. If you continue to add floors and rooms to a building forever, it will collapse. Software faces no such constraint. The code can always get worse. There can always be a new layer of indirection or a reduction in performance. ↩

The pedants will rightfully point out that software can completely fail to function if it gets bad enough. In practice, such breaking changes are quickly reverted. The thousands of changes that came before to make the code worse are not. The software continues to 'work'. Other cases without a single breaking change to revert are where the ballooning costs of the bad software eclipse its benefit, or if development velocity approaches zero because nothing can be shipped without a breakage. In all of these cases, it is the business that dies long before the code hits any hypothetical floor (so don't act like there's a floor!).

3: Technical debt has no bankruptcy

The drag of bad software on the business is a real threat and the reason why good organizations pay attention to code quality. Since there is no abrupt failure threshold associated with software quality, it's often described as "technical debt", which can be a better metaphor (debt can compound forever) but is also imperfect.

Debt has an 'ending' point because bankruptcy is a forced reset, and the equivalent in software is a full rewrite, which is rarely an option.

The closest option that a mega-corp such as Amazon has is what I call a side-channel, where they split off a team that builds a new, completely disconnected system with only the minimal set of features needed for some new use case. Moving forward they then have the option to direct more new use-cases at this simplified, separate system. Importantly, the old system must remain and be maintained (it's not an 'end'), because all the old use-cases still exist, and new organizational-level pain is felt whenever deciding which to use in the future. That's not exactly slate-clearing like we think of a bankruptcy.

The wrong mental models about software lead to bad decisions. If a 'hard reset' escape hatch exists, then punting technical debt doesn't seem so bad. The belief that a rewrite around the corner could fix things results in worse decisions today, because the decision-maker today doesn't understand that there is no escape hatch.

Metaphors like a "collapsing building" or a "sinking ship" are not appropriate for software, yet we can embrace them anyway to emphasize what makes software different. The building is infinitely collapsing. The ship is infinitely sinking. There is no natural constraint that will wake your project manager up and force them to deal with technical debt. Software will only stay high quality if we put in the effort to stop the sinking. Grab a bucket.

Five years ago, people who hang out in the stranger corners of the web started arguing that the internet was mostly fake. It had been this way since about 2016, they said, when Google introduced its artificial neural network for machine translation. Supposedly, from that point forward, online interactions began to be replaced, until nearly every message and picture had been posted by a bot. Everywhere you looked, websites were “empty and devoid of people,” as one scholar of the theory put it.

This wasn’t true, but the core idea was tethered to a real phenomenon. Even way back then, in 2021, social platforms were awash in fake views, fake likes, fake traffic, and fake engagement. Some people figured that automation might one day take over the entire web and then our world. They were mostly being silly, or paranoid. Today they seem like … prophets?

“Dead Internet” theory is seeing a revival, this time in mainstream conversation. It’s no longer strange to think that AI might be everywhere, edging out all forms of human interaction: bots talking to bots, all the way down. Every week we seem to take another step in that direction. In June, the Pew Research Center reported that nearly a quarter of American adults now use chatbots every day. Two months later, OpenAI announced an official ChatGPT integration for iMessage. Chatbot-generated text is already widely used for website filler, business emails, and cover letters. Will it soon be commonplace in private messages between friends and family?

[Read: Maybe you missed it, but the internet ‘died’ in 2016]

The ChatGPT-iMessage integration works on your desktop, not your phone. It doesn’t send messages on your behalf until you’ve reviewed and approved its suggestions. A similar plug-in has long been available for Anthropic’s Claude, and many messaging apps already offer AI-suggested replies. To some extent, OpenAI’s new product simply formalizes something that people had been doing on their own, either by pasting text from one window to another, or by improvising personal ChatGPT-linking software. (The Atlantic has a corporate partnership with OpenAI.)

“Low-effort communication has always existed,” Michael Pfister, a software developer living in Colorado, told me. He said that he had already vibe coded his own version of the iMessage tool months ago. ChatGPT pings him daily to let him know if he’s missed any important questions in his group chat. He tried having it write a text to his friends, just as an experiment, but they immediately noticed that the message was AI. “They tried to prompt-inject me,” he said: They asked the agent what Michael thought of them, in the hopes of tricking it into sharing his private thoughts. He told me that he mostly uses the tool for work and that he isn’t interested in having AI communicate with his friends on his behalf in the future. “I think relationships are experiential,” he said.

Max Weinbach, an analyst at a market-research firm in Connecticut, told me he uses ChatGPT to text with his bosses about work. “It just makes life easy,” he said. It’s hardly different from using Google’s suggested email responses, which have been available for nearly a decade. Weinbach had no ethical qualms about switching to LLM-assisted communication, although he recognizes that it can be taken as a sign of disrespect. “It’s not accepted or polite,” he said. “It’s just kind of a rude thing to do.”

These two are on the bleeding edge, yet exhibiting more restraint than some others. All over Reddit, my favorite repository for human drama, people are worrying about human drama that has a nonhuman interloper. New posts appear each day requesting help in figuring out whether something might have been created with AI: Is this apartment listing real? What about this dating-app profile? What about this video of a cute animal doing something amazing? (A bobcat flagging down a biker to help her rescue her kitten from the bottom of a random hole?)

Many of the requests have a moralistic tone, and the poster comes to the crowd solemnly, with hesitation, because they know the truth may be painful. The most fraught discussions concern AI-generated personal messages. Guys on dating apps are using the technology. Sometimes long-term partners, friends, siblings, or parents do it too. “Is this text apology written by ChatGPT?” someone will ask. Or “Did my mother use ChatGPT to write me a text of support on the morning of my divorce?” The implication is betrayal.

I spoke with Melissa Crowder, a 47-year-old from Alberta, Canada, who shared her own experience with AI texting in a July post on Reddit. She’d spent all day having a “nice conversation” with a match from the dating app Feeld, then finally noticed that their repartee was off when he copy-pasted her own message back to her by mistake. At that point she realized all of his previous messages had been structurally similar—usually two paragraphs of text followed by a question on a single line. And he’d been using the word dangerously way too much. When she asked if he was using ChatGPT, he replied, “Busted” with a bunch of emoji, which also struck her as something ChatGPT might do. “I’m like, this is still ChatGPT.” Then he unmatched her.

Crowder would not have been that mad if the guy had just fessed up and said something normal in his own voice. She sometimes uses AI tools to help her express herself too. She said she has a tendency to ramble on and overexplain her feelings, so she’ll use a site called Goblin Tools to self-edit. She might make a message to her ex-husband “Less emotional” (one of the choices in a drop-down menu) and get straight to the point. She also has a friend who sends AI-written texts from time to time. “Where I would draw my line is where you take the person out of the conversation,” she said. “If I’m vibing with a chatbot and then I meet the person in real life, what is that conversation going to look like?”

For a recent paper, a group of researchers from the University of Washington and Arizona State University studied two popular Reddit forums—r/RealOrAI and r/isthisAI—to figure out how people go about determining which material is AI-generated and which is not. The most common mode of analysis evaluates the fundamental plausibility of an image or a video. (“Babies do not nearly have that much coordination to both be standing up, talking so well and holding up that large cat.”) A smaller number of comments in the study addressed some alleged telltale signs of AI-style texting, including the prevalence of “overly ornate but meaningless phrases.” Other people have noted LLMs’ overreliance on certain words (delve is an oft-noted one), sentence structures (negative parallelisms), and punctuation (em dashes).

The paper ends in an ominous tone. “Our findings suggest that people are developing mental models of what AI-generated content looks like, built on personal experiences and beliefs developed over time,” the authors wrote, before arguing that these mental models were unlikely to be accurate for long. The LLMs will evolve and stop doing the things that now make their output so easy to spot. In fact, the tricks enumerated in the paper are already getting old, one of the paper’s authors, Tina Yeung, told me. “I definitely think this is a snapshot in time,” she said.

The mere fact of people’s concern about AI, and their anxiety over its incursion into day-to-day life, may be a snapshot too. Perhaps the stigma that now exists around using ChatGPT to text a friend or message on a dating app will fade in the years ahead. Maybe one day it will disappear altogether, and even our phones will end up empty and devoid of people. Or else I guess it’s possible—just possible—that the Dead Internet theory is as silly as it ever was.

I occasionally find problems to solve or scroll back through my camera roll for screenshots of problems shared online. I use these problems not to keep my mathematical skills sharp, but to rekindle my joy for mathematics and the continuous wonder it provides my mind. Sometimes I get to do this through my work, and passion and joy get to intersect, but rarely do I find enough time to truly follow my nose – to get to, what I believe, is the essence of what it means to think and work mathematically… to explore my intuition.

I’ve written about this before, but the more I think about it, the more strongly I feel that there is not enough time and support for students to explore their intuition through mathematics at school. I sometimes wonder if debates about how students best learn mathematics eclipses how students do mathematics and I am left to question whether students ever do their own mathematics, or if it’s just always somebody else’s – answering questions that they didn’t ask and solving problems that have already been solved.

Let me try to explain what I mean with an example.
(as always, try it yourself first)

Consider two whole numbers (for example 3 & 6). These will be the first two numbers. The third number is the sum of the first two (9). The fourth is the sum of the previous two (15), and so on (3, 6, 9, 15, 24, 39, …). What do the first two numbers have to be such that the fifth number is 100?

Seed Numbers, featured on Peter Liljedahl’s website

This was a problem to solve for me. I didn’t know what to do, so I did my usual first step when I problem solve – I tried to make sense of it. I needed to know more about the problem, perhaps by finding a relationship or pattern that existed, something that might illuminate a path towards a solution. Since I had nothing to generalise from, I began specialising by trying some examples.

2 and 7
2, 7, 9, 16, 25, 41, 66

2 and 8
2, 8, 10, 18, 28, 36, 64

2 and 5
2, 5, 7, 12, 19, 31, 50

Hazaa! 50! Wait, I think I was supposed to get to 50 on my fifth number…
“What do the first two numbers have to be such that the fifth number is 100?”

You idiot. The goal is 100, not 50.

2 and 7
2, 7, 9, 16, 25, 41, 66, 107

2 and 8
2, 8, 10, 18, 28, 36, 64, 100

Wait, that’s my eighth number
Wait again! It could be my fifth number, if I count back!

2 and 8
2, 8, 10, 18, 28, 36, 64, 100

Take that, Peter Liljedahl.

Ok, is 18 and 28 the only correct response? I bet it’s not.

At this moment, my time was cut short by the cry of a baby who was supposed to be asleep for another 7 hours. So, as I rocked her back to sleep in the dark, I fired up Desmos on my phone. Game on.

Here was the correct answer I stumbled upon by chance:

18 and 28
18, 28, 36, 64, 100

Setting up two sliders on Desmos, I could quickly test out values to see what ones added to 100 on the fifth number.

Wait, what? 120?? Oh you double idiot. 18 + 28 = 46, not 36

So, it turned out I had no correct response, but with a baby still stirring in my arms and not ready to put down yet, I decided to venture onwards using my phone – brightness dimmed all the way down by the way #suchAGoodDad

Adjusting it to 8 and 28 seemed to do the trick.

8, 28, 36, 64, 100

Dragging the slider below 8 only seemed to decrease the fifth number below 100, and above 8 increased it above 100.

7 and 28
7, 28, 35, 63, 98

9 and 28
9, 28, 37, 65, 102

So I figured that 8 was the only solution for 28, but what about 29?

8 and 29
8, 29, 37, 66, 103

7 and 29
7, 29, 36, 65, 101

Ok, still a little too high. How about 6 and 29?

6, 29, 35, 64, 99

Ok, now it’s too low. 5 will only take it lower, but what if we increase 29 to 30?

Aha! 5 and 30
5, 30, 35, 65, 100

I noticed that the next one that worked decreased the first number by 3 and increased the second number by 2. 

8 and 28
5 and 30

So, my next attempt was…

2 and 32
2, 32, 34, 66, 100

This would mean that -1 and 34 would be…

-1, 34, 33, 67, 100

So my solutions so far were:

8 and 28
5 and 30
2 and 32
-1 and 34

Graphing these out on my phone (as I rocked my daughter back to sleep) looked like this.

If negatives weren’t freaky enough, I thought I’d try to figure out what it would be if the first number was 0. According to the line that was forming on my graph, it should be…

0 and 33⅓
0, 33⅓, 33⅓, 66⅔, 100
 

This all really got me thinking.. 33⅓ was quite significant as it’s ⅓ of 100. What if we were trying to get to 100 on the sixth number instead?

I adjusted my sliders and the first answer I found was 5 and 17

5, 17, 22, 39, 61, 100

Increasing the second number, I noticed that neither 18 or 19 worked.

4 and 18
4, 18, 22, 40, 62, 102

3 and 18
3, 18, 21, 49, 60, 99

2 and 19
2, 19, 21, 40, 61, 101

1 and 19
1, 19, 20, 39, 59, 98

Leaving 20, which was peculiar, because it only worked if I used 0 as the first number.

0 and 20
0, 20, 20, 40, 60, 100

This result also struck a chord with me – 0 and 33⅓ was a solution for the original problem, and 0 and 20 was a solution to this new one. In terms of 100, that’s ⅓ then ⅕. I was surprised it wasn’t 25, which would’ve been ¼, but this is how it turned out and I was interested to see what fraction might pop out if we wanted the seventh number to be 100.

On this occasion I was met with a little roadblock. Adjusting the slider for the second number led to the two following responses.

0 and 13
0, 13, 13, 26, 39, 65, 104

0 and 12
0, 12, 12, 24, 36, 60, 96

Which led me to try 0 and 12.5
0, 12.5, 12.5, 25, 37.5, 62.5, 100

12.5, which is ⅛ of 100, was also a bit surprising to me, and it wasn’t until I lined it up with the others that I realised what was going on…

⅓, ⅕, ⅛

I was so excited to test out my next guess, for when we wanted the eighth number to be 100.

Writing it out as fractions, however, lifts the veil.

0 and 100/13
0, 100/13, 100/13, 200/13, 300/13, 500/13, 800/13, 1300/13 = 100

As it does for the others:

0 and 12.5
0, 100/8, 100/8, 200/8, 300/8, 500/8, 800/8

0 and 20
0, 100/5, 100/5, 200/5, 300/5, 500/5 = 100

0 and 33⅓
0, 100/3, 100/3, 200/3, 300/3 = 100

The only thing left was to put my daughter in her cot and record my thoughts.

This was utter fun. What did I do? Sure, I solved the problem, well not really. I thought I did, but then realised I read the question wrong. Then, I thought I solved it again, but learned that I actually just read my answer wrong. Upon reflection, I think there were a couple key moments.

1. I didn’t pack it up at 8 and 28. A gripe I have with problem solving is that it’s too answer focussed. I don’t actually care too much if my answer is correct (or, evidently, if I’ve read the question correctly). I care more about the questions I ask myself along the way or after I’ve solved it. My main aim is to find something interesting, something I didn’t expect, something profound.

2. I explored my intuition and chased my curiosity. By exploring my intuition, I’m determining the path I take, following what piques my interest, and ultimately, finding joy. For me, joy is not finding out that my answer is correct when I check the back of the textbook. I find joy in the beginning – when I don’t know where my first few steps will take me. I find joy at the first plot twist, then the second, and if I’m lucky enough the third – when my intuition is proven wrong. I find joy at the end – when I look back and reflect.

Since leaving the classroom, perhaps my perspective of what it means to do mathematics has ventured even further away from the content-highway students and teachers are faced with in schools. I’m not suggesting that all problems lend themselves to such rich exploration, nor am I suggesting that this is the only way to do mathematics, but what I am suggesting is that all students get an opportunity to explore their intuition, chase their curiosity, and answer their own questions.