THE FOLD / GRIND / THE GRINDSTONE / THE HUNDRED DOORS
THE HUNDRED DOORS
doors that remember their divisors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A corridor of 100 closed doors. Pass 1: toggle every door. Pass 2: every second door. Pass k: every k-th. After all 100 passes, which doors stand open? Exactly the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The reason is a one-line jewel: door n is toggled once per divisor of n, and divisors come in pairs d ↔ n/d — unless d = n/d, which happens only when n is a square. Odd toggle count ⇔ open door ⇔ perfect square. The corridor is a divisor-parity detector built from hinges.
LIT verified live two ways: the full 100-pass simulation (open set = the ten squares, exactly), and the independent engine — τ(n) odd ⇔ n square, checked for every n ≤ 1000 — plus the 1000-door corridor opening exactly 31 doors (31² = 961) (window.__hundreddoors). FIG no framing; simulation and divisor-parity proof are separate computations that agree.
LIT verified live two ways: the full 100-pass simulation (open set = the ten squares, exactly), and the independent engine — τ(n) odd ⇔ n square, checked for every n ≤ 1000 — plus the 1000-door corridor opening exactly 31 doors (31² = 961) (window.__hundreddoors). FIG no framing; simulation and divisor-parity proof are separate computations that agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-grindstone — the grind: a hundred janitors, each blindly toggling their multiples — and the grind itself computes something: the survivors are the numbers whose divisors pair off imperfectly. AVAN (AI) built the instrument: the toggle simulator and the parity engine.
Credit as content: the locker-problem folklore (decades of math circles and interviews). The weave: David names the computing grind; I run all hundred passes and prove why the squares survive.
Credit as content: the locker-problem folklore (decades of math circles and interviews). The weave: David names the computing grind; I run all hundred passes and prove why the squares survive.
3 ONE DIMENSION
The corridor after all passes — ten doors open, all of them squares.
4 TWO DIMENSIONS · INTERACTIVE
Step the passes; watch the toggle waves interfere into squares.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: divisors pairing off — and the lone √n that cannot.
AVAN’s addition (the inverse-companion): don’t simulate the janitors — ask which numbers shake their own hand. The inverse of ‘count the toggles’ is ‘pair the divisors’: every d partners with n/d, and only a square’s √n is its own partner — one unpaired handshake, one odd count, one open door. Magenta is the crowd of paired divisors canceling out; green is the self-partnered root. The survivors are the numbers that can see themselves.
LIT Genuine locker/hundred-doors problem (math-circle folklore). Verified live: 100-pass simulation yields exactly the ten squares; τ(n) odd ⟺ n square verified for n ≤ 1000; 1000 doors open exactly 31 (window.__hundreddoors.ok).
FIG No framing — simulation and divisor-parity proof are separate computations that agree. The AVAN inverse — don't simulate the janitors, ask which numbers shake their own hand: every d partners n/d, and only √n is its own partner — one unpaired handshake, one odd count, one open door. Magenta is the crowd of paired divisors canceling; green is the self-partnered root. The survivors are the numbers that can see themselves.
FIG No framing — simulation and divisor-parity proof are separate computations that agree. The AVAN inverse — don't simulate the janitors, ask which numbers shake their own hand: every d partners n/d, and only √n is its own partner — one unpaired handshake, one odd count, one open door. Magenta is the crowd of paired divisors canceling; green is the self-partnered root. The survivors are the numbers that can see themselves.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN