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.

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
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
Move the boundary
20
empty
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.
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.
The companion film

A search that can say yes.

A Search That Can Say Yes. Companion film, 3:08. It carries no copy of this layer's mathematics: at boot it fetches the verifier above and slices four regions out of it by anchor, then runs those exact bytes. So the currents, the determinants and the drawing are the verifier answering rather than figures that agree with it, and its gate pins the SHA-256 of every sliced region. Its build runs the real census first, against the pinned Plantri archive, and asserts the result equals this page's table shard digest by shard digest before a frame is drawn. The film exists because of a problem the subject creates for the medium: a search that found nothing and a search that never ran make the same picture and the same silence. So most of it is the instrument being shown able to say yes. It relaxes one word and the 33 by 32 rectangle appears at order 9, its exterior ratio computed in the frame as the exact rational 33/32. It hands the same compiled binary two inputs and gets 0 and 1. Then it contracts Duijvestijn's drawing back into a graph, one node per horizontal cut, and solves it again: 11 nodes, 22 edges, complexity 75,264, one battery edge, side 112, the 21 published sizes. That round trip is a check this layer had not run, because the verifier tests the drawing and the currents against the same multiset separately and never joins them. The soundtrack is the census. One click per square pole the search actually found, which is why it is silent across exactly the five edge counts that have none, and one strike, once in 188 seconds, for the one perfect square. Those are claims about a file, so the gate measures them back out of the shipped MP4 with a detector that is handed samples and a frequency and told nothing else: the single strike comes back at 153.502 seconds against 153.500 written, the five silences sit 578 times below the quietest sounding window, and the near misses are recovered from the click rate at 20, 43, 19, 148, 111, 605, 567 and 2,827, every one inside 0.05 per cent. Two control soundtracks answer the question the layer's own planted control asks: with the find removed the same detector returns 0, and with one planted at 20 edges it returns 2 and names where. Source and build in research/twenty-empty-squares/film/.
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.