n = 40 · Case Records · The Squares Project

n = 40 open ★=R

$\frac{67}{10} \leq s (40) \leq 4 + 2 \sqrt{2}$

The best packing known for 40 squares, side

4 + 2 \sqrt{2}

, Frits Göbel 1979

6.7006.828

678

6.3257.325

\sqrt{n}

\sqrt{n} + 1

Proven

$6.700000$ $\leq s (40) \leq$ $6.828428$

new result
exact
rigid (catalogue)

Citation record n-040

lowerwand125 after Tokoharu, Levy et al. 2026, GitHub (confirmed T-074)

upperGöbel 1979, Squares in Squares

Open

optimality

Bounds

Best known packing

$4 + 2 \sqrt{2}$

6.82842712474619

Found by: Frits Göbel 1979
Construction: hand, catalogue rigid
Source: [Kingbird]
Evidence: E-kingbird-upper-register, E-n040-gobel-upper

Verified upper bound

$4 + 2 \sqrt{2}$

6.82842712474619009760337744842

The reported value, verified here.

Evidence: E-n040-gobel-upper

Reported lower bound

$\frac{67}{10}$

Proved by: wand125 2026
Kind: counting
Scope: Unrestricted square packing with independent rotations and disjoint interiors.
Note: Rectangle-density certificate in Tokoharu's format, reported in the retained source and accepted there by Tokoharu's unchanged interval checker. The verified lane records the local replay separately.
Source: [wand125 rectangle bounds 2026-10-01]
Evidence: E-wand125-rectangle-2026-10-01-report

Verified lower bound

$\frac{67}{10}$

The reported value, verified here.

Evidence: E-wand125-rectangle-2026-10-01-source-replay

Gap

$2 \sqrt{2} - \frac{27}{10}$ ≈ 0.12842712…

Verified upper minus verified lower.

Results in the register

T-007 V0 C1 Nagamochi · 2026-08-31 · 321 cases
$s (n) \geq min (⌈ \sqrt{n} ⌉, \sqrt{n - 2 ⌊ \sqrt{n} ⌋ + 1} + 1)$ for $4 \leq n \leq 324$
T-013 V3 C3 Levy · 2026-08-31 · n = 40
Goebel's $n = 40$ packing: seven verified first-order flexes, each refused at second order
Claim and records
Claim
Goebel's $n = 40$ packing is infinitesimally flexible -- seven verified independent first-order flexes turn the sixteen-square tilted block -- and every retained flex is refused at second order by a verified non-negative self-stress, exactly over Q(sqrt 2), so no first-order argument can establish rigidity here.
Next rung
Characterise the admissible cone past dimension 45 (the corner-touching branch obstruction, D-391) or establish local rigidity by another route; the rigidity property stays undetermined until then.
Significance
The first separation of infinitesimal from finite rigidity in this literature's cases: the sources assert rigidity without the distinction, and this refines rather than contradicts them.
Novelty
apparently-novel Not found in the recorded search, subject to its stated gaps
Records
n = 40 register evidence 1
n = 40 register evidence 1
T-044 V3 C3 wand125 after Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 17 cases
Weighted point lower bounds for ten counts in $n = 26 \dots 72$ , plus seven from the same files
T-045 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 15 cases
Rectangle-density lower bounds replayed at 15 counts in $n = 18 \dots 78$
T-046 V0 C0 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-09-29 · 48 cases
Rectangle-density lower bounds reported for 48 counts in $n = 18 \dots 95$
T-058 V3 C3 wand125 after Tokoharu, Daniel · 2026-09-29 · 100 cases
Rectangle-certificate ceiling α·UB(n) proved for $n = 1$ ..100; B·UB(n) on 64 grid rows
T-068 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-01 · 34 cases
Rectangle-density lower bounds verified at 34 counts in $n = 19 \dots 95$
T-074 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 31 cases
Rectangle-density lower bounds replayed at 31 counts in $n = 19 \dots 95$
T-083 V3 C3 Karakuş · 2026-10-02 · 301 cases
$s (n) \geq 1 / 2 + \sqrt{n - ⌊ \sqrt{n} ⌋ + 1 / 4}$ for every nonsquare $8 \leq n \leq 324$
T-085 V3 C3 Karakuş; chelokot · 2026-10-02 · 315 cases
Nagamochi 2005, Lemma 1 is false for every container with $a > 3$ and $b > 2$

Verification

upper: replayed here; lower: replayed here

—

Rigidity

undetermined, verified, exact algebraic

Evidence: E-n040-first-order-flexibility

Scope

Assessed and not settled, on a first-party exact argument rather than on a screen miss. Over Q(sqrt 2) at Goebel's exact construction, the packing is infinitesimally FLEXIBLE: seven retained directions turn the sixteen squares of the tilted central block and leave the twenty-four axis-aligned ones fixed, each verified in the field. Each is then refused at second order by a verified non-negative self-stress. The property stays undetermined rather than moving to not-rigid because an infinitesimal flex is not a motion -- along every one of these the gaps curve shut at order t^2 -- so what is settled is that no first-order argument can establish rigidity here, not that the packing moves. The cone is bounded to dimension at most 45 by 75 certificates and is not characterised; the seven known directions span 6. Fixed side throughout. What a source says about this packing's rigidity is carried by reported_upper_bound.catalogue_rigid and is deliberately not restated here as a finding of ours.

Evidence and sources

11 evidence entries

E-n040-wand125-rect-67-sqverify-fast-replay, E-wand125-rectangle-2026-10-01-source-replay, E-wand125-rectangle-source-replay, E-wand125-rectangle-2026-10-01-report, E-wand125-rectangle-report, E-wand125-point-bounds-report, E-wand125-point-source-replay, E-kingbird-upper-register, E-nagamochi-lower, E-basic-grid-upper, E-green-ds7-theorem10-reported-lower

[wand125 rectangle bounds 2026-10-01] lower bound proof
[wand125 rectangle bounds 2026] lower bound proof
[wand125 point bounds 2026] lower bound proof
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey

In the frontier survey On GitHub

$s (40)$ — open

External intake, 2026-10-01. wand125’s rectangle-density source reports $s (40) \geq 67 / 10 = 6.7$ , with total mass $3999 / 100 = 39.99 < 40$ , accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again, after this repository’s exact audit checked that the regenerated checker input is the published one and checked the mass and net premises, so it is also verified. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-27. wand125’s rectangle-density source reports a direct $1339 / 200 = 6.695$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $3999 / 100 = 39.99 < 40$ , accepted by Tokoharu’s unchanged interval checker. This repository’s exact audit checks that the regenerated checker input is the published one, and checks the mass and net premises; the complete coverage replay has not yet run here, so the verified lower bound is unchanged. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-22. wand125’s retained point-certificate source reports $s (40) \geq 13 / 2 = 6.5$ , which was the verified lower bound until 2026-10-02. The complete exact replay verifies the retained certificate.

Open. The verified interval has endpoints $6.695$ and approximately $6.828427125$ , exactly $1339 / 200$ and $4 + 2 \sqrt{2}$ , leaving a gap of about $0.1334$ . Its lower end is wand125’s replayed rectangle-density certificate (T-045, 2026-10-02).

The packing

Found by Frits Göbel in 1979, via a hand construction. The catalogue flags it rigid: no continuous deformation is possible, so the contact conditions pin the side length exactly as an algebraic number.

The lower bound

The earlier record from [Friedman DS7] gives the lower-bound expression $2 \sqrt{2} + 8 \sqrt{5} / 5$ for $s (40)$ (approximately $6.406135888746$ ). Friedman’s DS7 survey, Theorem 10, k=6, reports this bound at n=40; reference [8] is Green’s private communication (2000). The source proof has not been recovered. That report remains as historical evidence; the verified point certificate gave the stronger bound $13 / 2$ , and the replayed rectangle-density certificate now gives $1339 / 200$ . The source audit compares the exact theorem expressions separately from opaque table decimals.

The exact replay verifies all five covering-certificate conditions over the complete rational direction net. A second interval implementation or generalized native importer would add confirmation and reuse, but it is not a missing premise of this fixed bound.

Corrected 2 October 2026: Nagamochi’s Lemma 1 is false (Karakuş 2026), so his closed form below is now a reported bound whose published proof is incomplete (review). This register had recorded that proof as verified, its own error, logged as defect D-516. Nagamochi’s earlier general closed form remains part of the evidence history and applies to every $N \geq 4$ :

s (N) \geq min {⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1}

Verification Code

The programs behind this case’s verified bounds, by their evidence. The code column says how the code that ran stands to the code its producer used. VERIFIERS.md says what each program is and whose it is.

bound	evidence	run	code	programs
verified lower	`E-wand125-rectangle-2026-10-01-source-replay`	replayed here	producer’s code	`V-tokoharu-verify-cpp` (external); `V-audit-wand125-rectangles` (first-party, premises)
verified upper	`E-n040-gobel-upper`	replayed here	independent	`V-sqpack-verify` (first-party)