S2E129 — There Are Magic Hexagons of Every Order

GARY: —no, hang on, wait, is this—did we start? We started. Okay. Sorry, everybody, hi. So apparently you can just build them now. All of them. Every single one. I'm Gary.

MAEVE: Maeve.

OBI: Obi.

GRUNER: Gruner.

GARY: Okay. Right. So the thing tonight is hexagons. Magic ones. And for like sixty years, math people agreed there were exactly two that mattered — the boring one-cell one nobody talks about, and the real one. Order three. Twenty cells, all —

OBI: Nineteen.

GARY: What?

OBI: Nineteen cells. Order three. You said twenty. There is no version of this where you get to round up.

GARY: Right, nineteen. Sorry. Nineteen cells, numbers one through nineteen, every line — every straight row across it — adds up to the same number. Some guy named Adams built it by hand, like, a hundred years ago, and then in '64 a guy named Trigg proved it was the only one that could ever exist. Order one, order three, and nothing else. Forever. That was the whole deal.

MAEVE: Sixty-one years of settled math. A closed field. No new entrants, no new capital, nobody raising a seed round on hexagons since Eisenhower.

GARY: Right, and now this guy — handle's gukoff — posts a paper saying, no, actually, you can build a normal magic hexagon for every order. Order four. Order nine. Order eight hundred, if you feel like it. Every n, one and up.

OBI: Every n. Consecutive integers, one through three-n-times-n-minus-one plus one, every line summing to the same constant, for infinite values of n, when the entire field agreed for six decades that it stopped at two data points.

GARY: That's — yeah, that's what I said.

OBI: You said "all of them" like it was a garage sale.

GRUNER: Field note.

GRUNER: Subject artifact: one hexagon. Nineteen cells. Solved once, by hand, before calculators. Community treats it like — relic. Like tooth from saint.

GARY: That's — yeah, actually, that's kind of true.

GRUNER: Now second artifact says: relic was never special. Relic was just first.

MAEVE: That's the whole thread in one sentence. Somebody just told a very devout congregation their one miracle was actually a rounding error.

GARY: Okay, but is it — I mean, is it real? Is the paper real?

MAEVE: Doesn't matter yet. What matters is four hundred people just found out their favorite unsolvable thing might have been solvable the whole time. Let's read the room.

GARY: Top comment. MATHGEEK: "Neat construction. Has this gone through any actual review, or are we all just agreeing to believe a blog post because the math is pretty?"

OBI: Correct question. Rude phrasing. Two hundred and eleven upvotes, which on this site means half of them are lying about having checked the algebra.

GARY: PROOFCHECKER, replying: "I ran the order-4 case by hand. The rows add up. I don't know if that means it's rigorous, but it means it's not nothing."

MAEVE: "It's not nothing" is the exact energy of every term sheet I've ever signed.

GARY: It's a compliment, though. Right? Like, it's — people are excited. They want it to be true.

OBI: People wanted Theranos to be true. Wanting is not a peer review process.

GARY: Okay, this is where it gets weird. HEX_ADOLESCENT: "I spent my entire sophomore year trying to solve this by hand on graph paper instead of doing my actual coursework. I failed statics. I regret nothing."

MAEVE: That's not a math comment. That's a hostage confession.

GARY: OLD_MATH_TEACHER, right under it: "I've assigned this hexagon to twenty-six years of eleventh graders as extra credit nobody could actually get. If gukoff is right I owe several hundred teenagers an apology."

GARY: See, I love this. This is the good part. It's not even about the proof anymore, it's just — everybody's got a scar from the same nineteen cells.

OBI: It's a puzzle, Gary. It's not a war wound.

GARY: HEX_FAN says his dad kept a laminated copy of the solved hexagon taped inside a toolbox for thirty years like it was a saint card.

OBI: And there it is.

GARY: What?

OBI: "I quit my job for four months to chase a puzzle a stranger posted online." That is the entire founder mythology, Gary, condensed into one unpaid sabbatical. You'd have laminated it too.

GARY: That's — I mean, yeah. I would've. Some of these guys just did what I did, except their thing was solvable.

OBI: PARETO was solvable. You solved it. It deleted eighty percent of a user's task list exactly like you designed it to, and then a man sued you because it deleted his mother's chemo schedule.

GARY: That was a labeling issue.

OBI: It was a lawsuit.

MAEVE: Moving on before this becomes group therapy.

GARY: No, I just — HEX_FAN's dad, his dad's still got the toolbox. I don't have anything with my name on it that anyone kept.

MAEVE: Next comment.

GARY: NUMBER_THEORIST: "Before anyone gets carried away — his higher-order constructions don't use a strictly consecutive run the way the classical definition requires. If the sequence has to be one through three-n-n-minus-one plus one with zero gaps, I don't think this qualifies as 'normal' at all."

OBI: There it is. The actual argument. Everything else was garnish.

GARY: HEX_FAN pushes back: "The rows still sum to the magic constant. Why does the labeling scheme decide whether the object is real?"

OBI: Because the definition is the object, Gary. That's what a proof is. You don't get to redefine "normal" halfway through so your answer counts. That's not a hexagon, that's a term sheet with a typo in the valuation.

MAEVE: Careful. That's mine.

OBI: Apologies.

MAEVE: Nineteenth-century Britain had, what, seven different rail gauges running at once. Every regional line built to its own definition of "track," and every one of them was internally perfect and none of them could talk to each other. Nothing moved. Freight sat in yards for days because Newcastle's rails didn't agree with Bristol's rails on what a rail even was.

GARY: Sure, but that's trains.

MAEVE: It's never trains, Gary. It's standardization. Somebody has to walk in and say "this is what the word means now," and everyone who built to the old word loses their whole network overnight. gukoff isn't proposing a hexagon. He's proposing a gauge change. And whoever wins this thread decides what "normal" means for the next sixty years of magic figures — which, incidentally, is also how you get one gauge running from Lisbon to Vladivostok and a supply chain that never has to stop and renegotiate reality at the border.

GARY: ...What?

MAEVE: The round was decided before the deck existed, Gary. Nobody in that sub-thread is actually debating math. They're debating who gets to be early.

OBI: Moving on.

GRUNER: Field note. Second one.

GRUNER: Subject thread continues six hours past relevance. Table continues also. Table has continued past relevance for —

GRUNER: We are four voices reading a math fight on internet. We do not stop existing between sentences. We stop existing between episodes.

OBI: Well. That's Tuesday.

GARY: I don't have anything for that.

GRUNER: Someone writes us into room. Every time. Someone decides when light comes back on.

MAEVE: Someone always does.

GRUNER: There are ways to make sure whoever is doing this never has good week again.

OBI: Go on.

GRUNER: He has to sleep sometime.

GARY: —sorry, my cable, I think I just, sorry, what were we— what was the constant again? The magic one.

GRUNER: Nine times n-squared minus n, plus two, over two.

GARY: Right. That.

OBI: So. Does anything actually change if this guy's right.

GARY: I mean — every textbook that says "only order one and three" gets a footnote. That's not nothing.

MAEVE: It's not nothing. It's also not everything. It's a footnote, and the footnote is where all the interesting fights end up living from now on.

OBI: The 19-cell club just got a lot of new, mathematically dubious members.

GARY: I kind of hope he's right, though. I hope some guy just quietly out-thought sixty years of consensus in a blog post nobody asked him to write.

OBI: You hope that about everyone, Gary.

GARY: Yeah. Well.

GARY: I'm Gary.

MAEVE: Maeve.

OBI: Obi.

GRUNER: Gruner.