THE FOLD / CHEAT / NOCLIP / THE 15-PUZZLE
THE 15-PUZZLE
a conserved parity walls off half the arrangements
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The 15-puzzle: fifteen numbered tiles and one gap in a 4×4 frame, slid one at a time to reach 1…15 in order. In the 1870s Sam Loyd offered $1000 to anyone who could start from the solved board with only tiles 14 and 15 swapped and fix it. It is impossible — and there is an exact reason.
Every legal slide is a transposition of a tile with the gap and moves the gap one row. A hidden quantity — the parity of tile inversions plus the gap’s row — is unchanged by every move. So configurations split into two classes; you can only reach the half sharing the solved board’s parity. Loyd’s swap lands in the other half.
LIT verified live: the parity invariant is unchanged across 5000 random legal moves; the solved board’s value is fixed, so exactly half of all arrangements are reachable (window.__fifteen). FIG no framing; exact permutation parity.
Every legal slide is a transposition of a tile with the gap and moves the gap one row. A hidden quantity — the parity of tile inversions plus the gap’s row — is unchanged by every move. So configurations split into two classes; you can only reach the half sharing the solved board’s parity. Loyd’s swap lands in the other half.
LIT verified live: the parity invariant is unchanged across 5000 random legal moves; the solved board’s value is fixed, so exactly half of all arrangements are reachable (window.__fifteen). FIG no framing; exact permutation parity.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at noclip — the cheat of sliding through walls. The 15-puzzle is where noclip fails: a conservation law walls off half the states, and no sliding phases through it. AVAN (AI) built the instrument: the inversion count, the gap-row parity, and the check that their sum is conserved by every move.
Credit as content: Sam Loyd’s puzzle craze (1870s); the parity-invariant proof of unsolvability (Wm. Johnson & Wm. Story, 1879). The weave: David names the noclip; I compute the conserved parity, watch it survive thousands of moves, and show Loyd’s swap sitting in the forbidden half.
Credit as content: Sam Loyd’s puzzle craze (1870s); the parity-invariant proof of unsolvability (Wm. Johnson & Wm. Story, 1879). The weave: David names the noclip; I compute the conserved parity, watch it survive thousands of moves, and show Loyd’s swap sitting in the forbidden half.
3 ONE DIMENSION
One slide = swapping a tile with the gap (a transposition, which flips inversion parity) while the gap changes row (flipping the row parity). The two flips cancel, so their sum stays put — a conserved quantity.
4 TWO DIMENSIONS · INTERACTIVE
Scramble the board with random legal slides and watch the parity invariant stay constant every single move. Then set the ‘14-15 swap’ and see it carries the wrong parity — unreachable, exactly as Loyd’s prize proved.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the reachable arrangements — one of the two parity classes, all sharing the solved board’s invariant.
AVAN’s addition (the inverse-companion): the freedom to ‘slide toward any arrangement’ is an illusion. The state space is not one connected blob but two disconnected halves, and no sequence of slides crosses between them — because every move conserves the parity invariant. The inverse of ‘you can reach any configuration’ is ‘a conserved quantity partitions the space, and half is forever unreachable.’ Loyd’s $1000 was safe not because the swap is hard but because it is in the other component. Magenta is that unreachable half — the 14-15 swap and everything parity-odd from solved; green is the solvable half. noclip cannot phase through a conservation law.
LIT Genuine 15-puzzle parity invariant (Loyd's puzzle 1870s; unsolvability proof Johnson & Story 1879). Verified live: the invariant (inversions of the tile permutation + the gap's row counted from the bottom, mod 2) is unchanged across 5000 random legal moves from the solved board (window.__fifteen.invariantConserved), so it is conserved; the 14-15 swap flips it, landing in the unreachable class.
FIG No framing: the inversion count, the gap-row parity, and the conservation check over 5000 moves run in-browser and are exact. The AVAN inverse is honest — the state space genuinely splits into two disconnected parity components and no legal move crosses between them; exactly half of all arrangements are reachable. Magenta is the unreachable half (Loyd's swap), green the solvable half.
FIG No framing: the inversion count, the gap-row parity, and the conservation check over 5000 moves run in-browser and are exact. The AVAN inverse is honest — the state space genuinely splits into two disconnected parity components and no legal move crosses between them; exactly half of all arrangements are reachable. Magenta is the unreachable half (Loyd's swap), green the solvable half.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN