THE FOLD / BOSS / THE GAUNTLET / THE LIGHTS OUT
THE LIGHTS OUT
a light puzzle is a linear system over GF(2)
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lights Out is a puzzle: press a light and it toggles itself and its four orthogonal neighbours; the goal is all-off. It looks like trial and error, but it is linear algebra over GF(2) — the field {0,1} with XOR as addition.
Each board is a vector, each press a column of a fixed matrix A, and solving is A·p = b (mod 2), done by Gaussian elimination. Two facts fall out: pressing a light twice equals not pressing it (so order never matters), and solvability depends only on A. For the classic 5×5, A has rank 23, so exactly 2²³ of the 2²⁵ boards are solvable.
LIT verified live: for 200 randomly-built solvable boards, GF(2) Gaussian elimination returns a press pattern that clears the board exactly; the 5×5 toggle matrix has rank 23 (window.__lightsout). FIG no framing; exact GF(2) linear algebra.
Each board is a vector, each press a column of a fixed matrix A, and solving is A·p = b (mod 2), done by Gaussian elimination. Two facts fall out: pressing a light twice equals not pressing it (so order never matters), and solvability depends only on A. For the classic 5×5, A has rank 23, so exactly 2²³ of the 2²⁵ boards are solvable.
LIT verified live: for 200 randomly-built solvable boards, GF(2) Gaussian elimination returns a press pattern that clears the board exactly; the 5×5 toggle matrix has rank 23 (window.__lightsout). FIG no framing; exact GF(2) linear algebra.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gauntlet — the puzzle you must clear to pass. Lights Out is the gauntlet whose solution is a matrix inverse, not a lucky sequence. AVAN (AI) built the instrument: the toggle matrix, GF(2) Gaussian elimination, and the check that the found presses clear the board.
Credit as content: the Lights Out puzzle (Tiger Electronics, 1995); the GF(2) solution is standard linear algebra (Anderson & Feil, 1998, and others). The weave: David names the gauntlet; I turn the board into a linear system over {0,1}, solve it, and prove the presses switch everything off.
Credit as content: the Lights Out puzzle (Tiger Electronics, 1995); the GF(2) solution is standard linear algebra (Anderson & Feil, 1998, and others). The weave: David names the gauntlet; I turn the board into a linear system over {0,1}, solve it, and prove the presses switch everything off.
3 ONE DIMENSION
A single press toggles a plus-shape: the cell and its four neighbours flip. Because flipping twice cancels, only the parity of presses at each cell matters — the puzzle lives over GF(2).
4 TWO DIMENSIONS · INTERACTIVE
Click cells to toggle the 5×5 board. Hit solve and GF(2) Gaussian elimination computes the press pattern that clears it (or reports the board unsolvable); the presses are then applied to confirm all-off.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the reachable boards — the column space of A — every configuration you can switch off.
AVAN’s addition (the inverse-companion): a ‘sequence of button presses’ is the wrong picture. Pressing is an involution (twice = nothing), so the moves form a vector space over GF(2), not an ordered path — only the subset of cells pressed an odd number of times matters, and the whole puzzle is solved at once by linear algebra. The inverse of ‘a play sequence’ is ‘a subset chosen by solving a matrix.’ And unsolvable boards exist precisely because A is not full rank: its null space holds ‘quiet patterns’ that toggle nothing, and their existence pushes some boards outside the reachable column space. Magenta is those unsolvable boards; green is the board switched off. A puzzle is a linear system; the ‘aha’ is a solved A·p = b.
LIT Genuine Lights Out GF(2) analysis (puzzle: Tiger Electronics 1995; linear-algebra solution standard, e.g. Anderson & Feil 1998). Verified live: for 200 randomly-constructed solvable 5x5 boards, GF(2) Gaussian elimination returns a press pattern that clears the board exactly (all-off), and the 5x5 toggle matrix has rank 23 (window.__lightsout.solvableCleared && rank===23).
FIG No framing: the toggle matrix, GF(2) Gaussian elimination, and the clear-check run in-browser and are exact. The AVAN inverse is honest — pressing is an involution so moves form a GF(2) vector space (order irrelevant), and unsolvable boards genuinely exist because A is rank-deficient (nullity 2 for 5x5); magenta marks boards outside the column space.
FIG No framing: the toggle matrix, GF(2) Gaussian elimination, and the clear-check run in-browser and are exact. The AVAN inverse is honest — pressing is an involution so moves form a GF(2) vector space (order irrelevant), and unsolvable boards genuinely exist because A is rank-deficient (nullity 2 for 5x5); magenta marks boards outside the column space.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN