Published positive control
One real square, waiting at 21
The c-net printed by A. J. W. Duijvestijn in 1978 has 22 edges. Exactly one deletion returns the order-21 square.
not run yet
Complete enumeration, bounded at order 20
Twenty empty squares
No simple perfect squared square exists at any order from 2 through 20. The complete c-net sweep returns an empty set. Move one step to order 21 and the same exact searcher returns a square.
ORDERS 2...20 · SIMPLE · PERFECT
The certificate
THE CLAIM
No simple perfect squared square exists at orders 2 through 20.
DOMAIN SWEPT
computing edge deletions from computing dual-oriented 3-connected simple planar c-nets with 6 through 21 edges. Orders 2 through 4 are discharged directly.
METHOD
Plantri 5.8 emits every graph class once. Every edge becomes the battery edge. Integer determinants test the square ratio, then exact currents test distinctness.
POSITIVE CONTROL
Duijvestijn's 22-edge c-net, order 21, reduced side 112.
NATIVE PLANTED CONTROL
The offline verifier adds a relabelled control c-net to the native census input. The unchanged checker finds exactly one hit; removing that input restores exactly zero.
RESULT
the empty set, exactly 0 survivors through order 20.
BOUND
The null stops after order 20. At order 21 a simple perfect squared square exists. Compound dissections are outside this claim.
The build-time census is recorded below. Run either live control.
Two ways the search can succeed
An empty answer needs a working instrument.
Both live controls use the browser's same unmodified graph-to-current search. It removes each edge, solves Kirchhoff's equations with reduced BigInt fractions, and accepts only nonzero, pairwise-distinct currents with exterior conductance exactly 1. The offline planted control separately exercises the native census.
Browser harness probe
Empty, inject, found, remove
This live differential proves only that the browser solver and report path respond to inserted data. It is not evidence about the native census and not a second discovery. The offline verifier performs the genuine plant through the native input.
records 0 · survivors 0 · empty set
The first survivor
Still nothing.
Order 20 means deleting one edge from every 21-edge c-net. The exact census found no perfect square.
result set = ∅
Relax one word
Square is the hard boundary.
Keep the nine component squares unequal and keep the dissection simple, but let the exterior be a rectangle. The desert ends immediately at order 9.
The exact area check is ready.
Why graphs cover every case
The tiling becomes a circuit.
Horizontal cuts become vertices.
Each component square becomes a unit-resistance edge. Its side length is the absolute current on that edge. Simple means no proper subset of at least two component squares has a union that is a rectangle. Joining the two poles turns such a dissection into a 3-connected planar c-net.
Every possible pole choice is tried.
Removing each c-net edge in turn covers every normal polar net. Duals only repeat a dissection after a quarter-turn, but this build deliberately retains those duplicates so completeness does not depend on dual canonicalization.
G = one c-net
e = candidate battery edge
H = G minus e
square iff τ(G) = 2τ(H)
perfect iff every current is
nonzero and pairwise unequal
τ is an integer determinant.
No floating point tolerance enters.
e = candidate battery edge
H = G minus e
square iff τ(G) = 2τ(H)
perfect iff every current is
nonzero and pairwise unequal
τ is an integer determinant.
No floating point tolerance enters.
The check
What ran, and what did not.
RECOMPUTED AT BUILD TIMEPlantri 5.8 regenerated the complete dual-oriented c-net census through 22 edges. A separate native exact-current checker tested every edge deletion, finding zero perfect squares through 21 edges and exactly one published-current hit at 22 edges.
PINNED INPUTOfficial archive SHA-256: e78a944116fec9f2c9f5e484206276cc2b0043bae803e9815f4b2683614629b8.
LIVE IN THIS PAGEThe 22 deletion positive control, a browser-only differential harness probe, the side-112 drawing, and the 33 × 32 rectangle area check.
NOT LIVE IN THIS PAGEThe 2,422,071-case census and genuine native plant are too large or platform-specific for a polite page load. The verifier recompiles Plantri, repeats the sweep, then passes the plant through the same native checker.
FREE CHOICESWe retain graph and planar-dual duplicates. The displayed order range begins at 9 for legibility, while the certified claim begins at 2.
UNCERTAINTY AND LIMITCompound squared squares are not covered. The page does not claim nonexistence after order 20. It does not redistribute paper scans or catalogues.
DIRECT LOW ORDERSOrders 2 through 4 cannot be perfect square dissections: the required interior cuts force repeated side lengths. The graph census begins with the six-edge K4 c-net.
REFUSAL PATHThe instrument refuses fractional orders, orders outside 9 through 21, and any boundary rule other than square or rectangle. The offline verifier asserts all refusals.
Primary sources
The record behind the empty set.
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, 1962Definitions, c-net correspondence, exhaustive generation method, exact current calculation, and the historical low-order null.
A. J. W. Duijvestijn, Simple Perfect Squared Square of Lowest Order, 1978The unique order-21 square, its polar net, Bouwkamp code, complexity 75264, reduction factor 336, and reduced side 112.
G. Brinkmann, H. Van den Camp, and B. D. McKay, Plantri 5.8Official isomorph-free planar graph generator, version and licence page. The build retrieved the official 2026-03-04 archive.
G. Brinkmann and B. D. McKay, Fast Generation of Planar Graphs, 2007The generation algorithm and validation context for Plantri.