Thinking About Thinking

For nearly four thousand years, puzzles tested minds. In the middle of the twentieth century, the enterprise turned around and began studying the thing it had been testing. The question of the new age was audacious and practical at once: what is solving? Is the knack a gift, or a craft with parts — nameable, teachable, trainable parts? Within a single decade, four very different answers appeared: a professor's checklist, an engineer's card-file, a columnist's stage, and — a little later — a logician's tavern. Between them they built the modern culture of thinking about thinking — the century in which "heuristic" left the seminar room, "lateral" acquired its modern meaning, and the puzzle column became a career — and this chapter follows that culture all the way to its strange present tense, where the oldest kind of riddle has returned to duty at the gate.

The Checklist

In 1945, the Hungarian-born mathematician George Pólya published a slim book with an immodest title: How to Solve It. Its claim was heresy against the romance of genius: solving is not lightning from nowhere. It has phases, and they can be written down. Understand the problem. Devise a plan. Carry it out. Look back. Under each phase, named tools — work backward from the goal; solve a simpler problem first; generalize; ask what is unknown, what is given, what connects them.

Written down like that, it looks almost too plain to matter. It mattered enormously. Pólya had done for problem-solving what Alcuin's title had only gestured at eleven centuries earlier: he made the method the explicit subject, separable from any particular problem and therefore teachable. The checklist is not an algorithm — nothing in it computes an answer — but it is a discipline aimed squarely at the two great amateur failure modes: diving in before understanding, and stopping the moment the answer appears. Against the first, Pólya sets the humble interrogation every solver in this book performs on instinct: what is unknown? what is given? what connects them? Against the second, his most underrated instruction, the fourth: look back. The problem you have just solved is a specimen; mine it — for the trick that generalizes, for the check you skipped, for the shorter road you missed — or you have only won, not learned.

Every named-heuristics tradition since descends from that little volume, including the Moves running through this book. Chapter 8 takes Pólya into the arena and asks what champions had to add to make the checklist survive contact with a ticking clock — because knowing the phases, it turns out, is not the same as commanding them under pressure.

The Card-File

At almost the same moment, on the far side of the descending Iron Curtain, a Soviet patent engineer named Genrich Altshuller began reading. Starting in 1946, he worked through the patent archive — some two hundred thousand inventions, eventually — looking not at what each invention did, but at what kind of trick it pulled. His heresy was the mirror image of Pólya's: inventive insight is not even personal. Across unrelated industries, the same contradictions recur (stronger but lighter; faster but more precise), and they are resolved by the same few dozen maneuvers — segmentation, inversion, doing it in advance. Altshuller distilled roughly forty such inventive principles, arranged a matrix pairing recurring contradictions with the principles that dissolve them, and published the first paper on the method, with Rafael Shapiro, in 1956. The system became known as TRIZ, the Russian acronym for the theory of inventive problem-solving.

TRIZ's unsettling gift is captured in one sentence: someone, somewhere, has already solved the abstract form of your contradiction. Insight as lookup, not lightning. The romance drains out; a card-file takes its place; engineers mine it to this day. Between Pólya and Altshuller, the mid-century message was clear — the aha could be systematized. What it could not yet be was popular. That took a magician's touch.

The Stage

In December 1956, Scientific American ran a one-off article on hexaflexagons — folded paper strips that hide and reveal faces as they flex — by a Tulsa-born magazine journalist named Martin Gardner, a man with, famously, no mathematics degree at all. Readers wanted more. From January 1957, "Mathematical Games" ran monthly, and it ran for a quarter of a century, to 1981 — some three hundred columns that became, with no real competition, the greatest amplifier in the history of recreational thought.

The column worked because it was a bridge with traffic in both directions. Downward, the frontier came to breakfast: Gardner introduced the world to John Conway's Game of Life (October 1970), to Roger Penrose's aperiodic tiles (January 1977), and to RSA public-key cryptography (August 1977) — in a games column. Upward went the amateurs: readers' letters sharpened problems, found errors, occasionally made discoveries; a generation of working mathematicians would later date their vocation to a specific month of "Mathematical Games," and his readers still convene as the Gathering 4 Gardner, running since 1993. Gardner also performed an act of historical justice with direct bearing on the previous chapter: he revived Loyd and Dudeney for a new century — and reported their provenance honestly. His creed, stated many ways across twenty-five years, was that the line between play and mathematics is imaginary.

When he retired, Douglas Hofstadter's successor column bowed to him in its very name: "Metamagical Themas," an anagram of "Mathematical Games." It is hard to name another figure who created more solvers. The amateur renaissance of the late twentieth century — the math-circle kids, the puzzle collectors, the recreational-mathematics shelf at the bookstore — is, to a first approximation, Gardner's column metastasizing.

Two Prophets of Play

The popular age produced its own theorists, and honesty requires grading them differently.

In 1967, Edward de Bono, a Maltese-born physician, published The Use of Lateral Thinking and gave everyday language a term it clearly needed: the sideways move, the deliberate escape from the groove of straight-ahead ("vertical") reasoning. His image was the hole: when you are digging in the wrong place, digging deeper does not help — the new hole must be started elsewhere. He built tools for forcing the sideways step — the provocation word PO (1972) and, most durably, Six Thinking Hats (1985), a meeting protocol in which a group wears one mode at a time (facts, feelings, caution, benefits, creativity, process) so that people stop defending identities and start taking turns at perspectives. The honest ledger, plainly stated: the empire was commercially vast, and the controlled evidence for a general, transferable "lateral thinking skill" stayed thin ⚠. The drills have real uses; the Six Hats survive as a genuinely clever de-ego-ing device; the brand oversold the science. Keep the exercises, grade the claims — a distinction this book will need again in Part V.

Raymond Smullyan needs no such asterisk. A logician who had worked as a stage magician, Smullyan published What Is the Name of This Book? in 1978 — a title that is itself a small self-referential trap — and turned a scrap of folklore — an island of knights who always tell the truth and knaves who always lie — into a systematic art form: self-reference, domesticated. The book is a ladder: each tavern riddle a rung slightly higher, until the reader who climbed for fun stands, without ever sitting through a lecture, at the edge of Gödel's incompleteness theorems — which To Mock a Mockingbird (1985) and Forever Undecided (1987) then walk right up to. Smullyan proved something lovely about the species: the deepest results in logic can be walked to, one puzzle at a time. The gymnasium had acquired a staircase to its roof. You will climb the lower rungs yourself in Chapter 5.

Between them, the two prophets mark the era's poles. De Bono preached breadth — escape the frame, whatever the frame. Smullyan practiced depth — follow one frame down to bedrock. The complete solver, this book will keep insisting, needs both hands.

The Object

In the spring of 1974, in Budapest, a young design lecturer named Ernő Rubik built a cube whose faces turned: fifty-four colored squares on six faces, each face free to rotate, the whole object somehow never falling apart. Hungarian patent, 1975; world launch, 1980; and then the craze of 1980–82, the largest the genre had seen since the winter of 1880 — this time with the mathematics visible just beneath the plastic. The cube also changed the sociology of being stuck: for the first time, a puzzle's difficulty was carried in public. A scrambled cube on a desk is a legible fact about its owner, and for two years much of the planet walked around holding its unsolved state in its hands.

The cube has 43,252,003,274,489,856,000 positions. Against that number, two facts define the object's meaning. First: in 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge proved — with about thirty-five CPU-years of computation — that every one of those positions is solvable in at most 20 moves. "God's number," the cubers call it. Second: no human method comes close. People solve the cube in fifty or sixty moves, by building small reusable tools — do-and-undo move sequences (the group theorist's commutators) that change a little while conserving the rest. The gap between 20 and 60 is the cube's real lesson, and it is one of the permanent lessons of this book: knowing a solution exists is not the same as owning it. Existence and findability are different currencies, and almost everything in Part III is about earning the second.

The Grid

Before the next relay race, notice a pattern the whole lineage obeys: every puzzle boom in this book rides a media technology. Papyrus carried Ahmes; court manuscripts carried Alcuin; cheap newsprint carried the 15 Puzzle and the crossword; the glossy monthly carried Gardner; molded plastic carried Rubik. The remaining booms of this chapter ride the web and the smartphone, and the pattern is worth keeping, because it predicts: whatever the next medium is, a puzzle will be among the first things it learns to do.

The same decades democratized the logic puzzle by way of an unlikely relay race. In 1979, Howard Garns, a retired American architect, published a modest number-placement puzzle called "Number Place" in a Dell magazine. It emigrated to Japan, where the publisher Nikoli refined it and, in 1984, named it — sudoku, "the digits must be single." It simmered there for twenty years until Wayne Gould, a retired judge who had spent years writing a computer program to generate the puzzles, sold The Times of London on the idea; the first grid ran on 12 November 2004, and by 2005 the boom was global. The Lo Shu's grandchild — you met the ancestor two chapters ago — had become the daily meditation of hundreds of millions: constraint propagation as a breakfast ritual, no arithmetic required, every deduction forced.

The mathematics, characteristically, arrived after the masses. There are 6.67×10²¹ valid completed grids (Felgenhauer and Jarvis, 2005). The minimum number of clues that can force a unique solution is exactly 17 (McGuire, Tugemann, and Civario, 2012 — by exhaustive computer search). Computer science adopted the puzzle as a laboratory; Donald Knuth's "dancing links" made it a standard testbed for exact-cover algorithms. And a Tokyo mathematics teacher, Tetsuya Miyamoto, pushed the pedagogy one step further in 2004 with KenKen, built on his classroom philosophy — "the art of teaching without teaching": design the constraints well enough and the student instructs herself.

The Curriculum and the Room

Two smaller inventions of the era deserve their paragraphs, because each moved the gymnasium somewhere new.

In 2001, Colin Hughes launched Project Euler — the name saluting the master of Königsberg, whom you last saw in a one-sentence signpost in Chapter 1 — a numbered sequence of mathematical problems designed so that brute arithmetic fails and only algorithmic insight — the right representation, the right theorem, the right complexity class — runs in under a minute. That one-minute rule is a quiet piece of setter's genius: it draws a bright line between having an answer and having an idea, and it grades the difference automatically. Mathematics and programming, fused into a single self-paced curriculum. The puzzle had become a syllabus.

In 2007, in Kyoto, Takao Kato of the company SCRAP staged the first Real Escape Game — a room you are locked in, which is itself the puzzle — inspired by point-and-click browser games and built of physical props and hidden mechanisms. Europe's first dedicated venue, Parapark, opened in Budapest (that city again) in 2011, and a global boom followed in 2013–15. What escape rooms added was everything print had to leave out: the body and the team. Search with your hands. Divide the roles. Reason under a clock, out loud, with friends. Anyone who has run one knows what the format really measures — not cleverness alone but the meta-skills around it: who searches, who synthesizes, who watches the time, who says out loud that the group has now spent nine minutes on the same locked cabinet. Print never tested that. Rooms do. Puzzling, solitary since Ahmes, had become architecture — and a social organism.

The Mirror

The lineage's newest mass ritual began, fittingly, as a love token. In October 2021, the software engineer Josh Wardle released a small web game he had built for his partner, Palak Shah: one five-letter word a day, six guesses, tiles turning green, yellow, or gray. Wordle's masterstroke arrived that December — a share button that posts your grid of colored squares without the letters: proof of work, spoiler-free. The grids flooded social feeds; by January 2022 the New York Times had bought the game for a price reported as "low seven figures."

Wordle's design lessons are a compressed course in why puzzles catch: scarcity (one a day — you cannot binge), community (everyone faces the same puzzle), and verification-without-spoilers (the grid shows that you solved, and how narrowly, never what). But its loveliest effect was epistemic. Within weeks, ordinary players were arguing about optimal opening words — which guess extracts the most information from the hidden answer? — and the arguments were, whether the arguers knew it or not, entropy calculations, made fully explicit in a widely watched 3Blue1Brown analysis (February 2022). A planet practicing information theory before breakfast. Dyson's wartime common room, scaled to a species.

The Gate Reopens

Which brings the lineage home.

In November 2019, the AI researcher François Chollet published a paper titled "On the Measure of Intelligence," and with it a test called the Abstraction and Reasoning Corpus — ARC. Its format is almost childishly humble: small grids of colored cells; a few examples showing a transformation, before and after; deduce the rule and apply it to a fresh case. Each task's rule is novel — invented so that no library of memorized patterns will contain it — and the rules lean only on the kind of knowledge a five-year-old owns: objects persist, shapes repeat, things fall and fill and mirror. Shown three examples, a person very often just sees it, and then struggles to say what exactly was seen. That gap — instant for the eye, awkward for the tongue — is the test's whole cunning.

Chollet's argument for this austerity cuts at the deepest assumption of the machine-intelligence age: skill is not intelligence. Any sufficiently trained system can be skilled at a task it has seen. Intelligence, he argued, is the efficiency of acquiring new skill — what you do when the rule is genuinely new. So the honest test is few-shot rule induction: tasks easy for humans and, by design, hard for machines.

Read that description again slowly. One question. Engineered so that a certain kind of mind slides straight off it. Posed at a gate everyone now wants to pass. Everything staked on the answer.

The lineage that opened at Thebes has closed its great circle: the gate-keeping riddle is back, and this time it is pointed at machines. Every gate asks three questions at once — who passes, who decides what passing means, and what passing actually proves — and the third has become, quite suddenly, one of the live intellectual questions of the age. Which minds pass, which slide off, how fast the scores are moving and what they mean — that is Part IV's story, told there with numbers and dates, because in that arena the numbers will not sit still. The Sphinx, at least, would find the whole arrangement familiar: a new kind of traveler on the road, an old kind of question in the way.

Before the scoreboard, though, the armory. Whatever species ends up reading it, the Moves are worth owning — and the next four chapters put them, one at a time, into your hands.