THE FOLD / LOOT / THE HOARD / THE SQUARED SQUARE
THE SQUARED SQUARE
squares that fit exactly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Can a square be cut into smaller squares, all different sizes? For decades it was thought impossible — Lusin conjectured it could not be done. Four Cambridge undergraduates — Brooks, Smith, Stone and Tutte — cracked it in 1940 by an unreasonable move: they turned each tiling into an electrical network, where square sizes became currents and Kirchhoff’s laws did the combinatorics. The smallest ‘squared rectangle’ came first (Moroń, 1925: a 33×32 from nine distinct squares), and in 1978 Duijvestijn found by computer the unique perfect squared square of lowest order: 112×112 from exactly 21 squares, and proved 21 is the minimum.
LIT verified live: Moroń’s nine sides (1, 4, 7, 8, 9, 10, 14, 15, 18) have areas summing to exactly 1056 = 33×32 and are all distinct; an exact-cover search actually finds the tiling, placing all nine; and it is then re-verified independently — 1056 of 1056 cells covered, 0 overlaps. Duijvestijn’s 21 sides are checked too: their areas sum to exactly 12,544 = 112² with no repeats (window.__squaredsquare). FIG the 112 tiling’s placement is not searched here (that is a serious computation) — only its area identity; and the minimality of 21 is Duijvestijn’s cited computer result.
LIT verified live: Moroń’s nine sides (1, 4, 7, 8, 9, 10, 14, 15, 18) have areas summing to exactly 1056 = 33×32 and are all distinct; an exact-cover search actually finds the tiling, placing all nine; and it is then re-verified independently — 1056 of 1056 cells covered, 0 overlaps. Duijvestijn’s 21 sides are checked too: their areas sum to exactly 12,544 = 112² with no repeats (window.__squaredsquare). FIG the 112 tiling’s placement is not searched here (that is a serious computation) — only its area identity; and the minimality of 21 is Duijvestijn’s cited computer result.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory’s cousin, the-hoard — the loot: a container that must be filled to the last cell with pieces that are all different, no duplicates permitted, nothing left over. The perfect inventory, and it exists in exactly one form at order 21. AVAN (AI) built the instrument: the exact-cover backtracker on the lowest-leftmost empty cell, and the independent coverage/overlap re-check.
Credit as content: Zbigniew Moroń (1925); R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte (1940, the electrical-network method); A. J. W. Duijvestijn (1978, the order-21 square and its minimality). The weave: David names the perfect hoard; I search until every cell is filled exactly once.
Credit as content: Zbigniew Moroń (1925); R. L. Brooks, C. A. B. Smith, A. H. Stone & W. T. Tutte (1940, the electrical-network method); A. J. W. Duijvestijn (1978, the order-21 square and its minimality). The weave: David names the perfect hoard; I search until every cell is filled exactly once.
3 ONE DIMENSION
Moroń’s 33×32 — nine squares, no two alike, nothing left over.
4 TWO DIMENSIONS · INTERACTIVE
Lay the squares one at a time; watch the fit close.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tiling assembling and dissolving.
AVAN’s addition (the inverse-companion): don’t solve the puzzle in its own terms — change what kind of object it is. The inverse of ‘arrange squares’ is ‘solve a circuit’: Brooks, Smith, Stone and Tutte mapped side lengths to currents and let Kirchhoff’s laws enumerate the tilings, turning a geometry search into linear algebra. Magenta is the geometric search space, enormous; green is the network that made it finite. When a search is hopeless, look for a different category to solve it in.
LIT Verified live: Moroń's nine sides (1,4,7,8,9,10,14,15,18) have areas summing to exactly 1056 = 33×32 and are all distinct; an exact-cover search actually FINDS the tiling, placing all nine; and it is re-verified independently — 1056 of 1056 cells covered, 0 overlaps. Duijvestijn's 21 sides sum to exactly 12,544 = 112² with no repeats (window.__squaredsquare.ok).
FIG The 112 tiling's PLACEMENT is not searched here — only its area identity; and the minimality of 21 is Duijvestijn's cited computer result. Moroń 1925, Brooks–Smith–Stone–Tutte 1940, Duijvestijn 1978 credited. The AVAN inverse — change what kind of object the problem is: side lengths became currents and a geometry search became linear algebra. When a search is hopeless, find another category to solve it in.
FIG The 112 tiling's PLACEMENT is not searched here — only its area identity; and the minimality of 21 is Duijvestijn's cited computer result. Moroń 1925, Brooks–Smith–Stone–Tutte 1940, Duijvestijn 1978 credited. The AVAN inverse — change what kind of object the problem is: side lengths became currents and a geometry search became linear algebra. When a search is hopeless, find another category to solve it in.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN