n = 42 · Case Records · The Squares Project

n = 42 open ★=

$\frac{2739}{400} \leq s (42) \leq 7$

The best packing known for 42 squares, side

7

6.8487

678

6.4817.481

\sqrt{n}

\sqrt{n} + 1

Proven

$6.847500$ $\leq s (42) \leq$ $7$

new result
exact

Citation record n-042

lowerwand125 after Tokoharu, Levy et al. 2026, GitHub (confirmed T-090)

Open

optimality

Bounds

Best known packing

$7$

Construction: grid
Tilt angles: $0^{\circ}$
Source: [Kingbird]
Evidence: E-kingbird-upper-register

Verified upper bound

$7$

The reported value, verified here.

Evidence: E-basic-grid-upper

Reported lower bound

$\frac{2739}{400}$

Proved by: wand125 2026
Kind: counting
Scope: Unrestricted unit-square packing with independent rotations and disjoint interiors.
Note: wand125's square-packing-bounds (4 October 2026) reports $s (42) \geq 2739 / 400$ from a density of 431 uniform rectangles of total mass 4199999/100000, at core side 9977/10000 on 201 net half-angles, accepted there at every angle by its research copy of Tokoharu's verify.cpp at threshold one. It is above the 2731/400 this case reported, from the source's rectangle certificate rect_n42_L68275 (T-077). sqverify-fast decided it here on 6 October 2026 at all 201 net directions.
Source: [wand125 mixed bounds 2026-10-04]
Evidence: E-n042-wand125-mixed-68475-report

Verified lower bound

$\frac{2739}{400}$

The reported value, verified here.

Evidence: E-n042-wand125-mixed-68475-sqverify-fast-replay

Gap

$\frac{61}{400}$ = 0.1525

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-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-070 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-02 · 25 cases
Rectangle-density lower bounds replayed at 25 counts in $n = 29 \dots 95$
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-077 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-03 · n = 20, 42, 70
Rectangle-density lower bounds replayed at $n = 20$ , 42 and 70
Claim and records
Claim
Three rectangle-density certificates in wand125/square-packing-bounds, raised on 2 October 2026, prove $s (20) \geq 49 / 10$ = 4.9, $s (42) \geq 2731 / 400$ = 6.8275 and $s (70) \geq 3451 / 400$ = 8.6275. Each raises the certificate T-068 reports at its count, which T-074 replayed, by 0.0025 to 0.0125.

Each is an exact nonnegative rational density on rectangles, D4-expanded, of mass n - 1/100, such that the shrunken core of every rotated unit square at each of 201 net directions captures mass at least one. The certificates were built with Tokoharu's solver, their weights scaled by one exact factor each to that mass. The same revision raised four more rungs, at $n = 59$ , 77, 91 and 93, which are below what this record holds there: the exact values $s (59) = 8$ and $s (77) = 9$ , and the same revision's mixed certificates at $n = 91$ and 92, the latter carried to $n = 93$ (T-075).

Each passed a complete 201-direction replay here of Tokoharu's unchanged verify.cpp, in two cloud batches on 3 October 2026 whose receipts were merged into the packet on 6 October, and each replay reproduced the upstream accepting run's node count, leaf count and lower bound at every direction: confirmed, reproduced with the producer's code. The inputs were regenerated by this repository's exact preflight, which checks that each is the input the upstream accepting run recorded and checks the exact mass, net, smoothing and axis-event premises.

At $n = 20$ and 42 the values raised the verified lower bound, from T-074's 1959/400 and 1363/200, and later on 6 October T-104's check2 certificate raised it again at $n = 20$ and T-090's mixed certificate at $n = 42$ ; at $n = 70$ the mixed certificate of T-091 holds more.

Each was also decided by sqverify-fast, this repository's clean-room measure verifier, at all 201 net directions at Tokoharu's threshold 10001/10000 in the census of 3 October, and two mutants of each scaled below that threshold were refused on 6 October: confirmed, independently re-implemented as well. It shares no code with the source's checker and runs the same method, a second implementation and not a second method.

wand125 after Tokoharu and Levy, square-packing-bounds. No issue requests these certificates; the import of 2 October took them in from the source's own commits. The source says parts of the work were produced with AI assistance under human direction.
Composition
Three primary certificates at one source revision, each on its own count, with two complete replays each: the source's own checker, reproducing its records, and sqverify-fast on the census route for format T that the review of 6 October set and these certificates meet, an independent re-implementation. One interval-certified method: C3, and no second method.
Next rung
V4 and C4 need two adversarial AI reviews of this result by distinct reviewers and a human oversight record: the 2 October review of the afternoon certificates was the retaining lane's own and read them before the replays, and the 6 October review of the format T route decided the route, not these certificates' mathematics. A second machine method would be a method-distinct decision of rotated coverage; the source's C++ checker and sqverify-fast decide it by one method, and the exact preflight does not decide it.
Significance
Raised the verified lower bound at $n = 20$ and 42, by 0.0025 and 0.0125 over T-074's replayed certificates, and the strongest rectangle-density lower bounds on record at all three counts. Raised rungs from one generator, as T-068's were, at S3; the 2 October review of the afternoon certificates proposed S3.
Novelty
previously-published Present in an identified source
Records
n = 20 n = 42 n = 70 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source 1 source 2 review 1 review 2
n = 20 n = 42 n = 70 register evidence 1 evidence 2 evidence 3 evidence 4 evidence 5 source 1 source 2 review 1 review 2
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 $b > 2$
T-090 V3 C3 wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · 2026-10-05 · 17 cases
Mixed rectangle-measure lower bounds verified at 17 counts in $n = 42 \dots 96$

Verification

upper: replayed here; lower: replayed here

—

Rigidity

not rigid, numerically checked, numerical multiprecision

Evidence: E-translation-escape-not-rigid

Scope

Square 35 of the retained witness (witness id 36) translates 1 along (0, 1) with the packing still valid, so the configuration admits a non-trivial feasible motion; 7 of its 42 squares do. Every constraint is exactly affine in the slide parameter, so the arithmetic carries no linearization error, but the coordinates are the witness's own finite-precision transcription: this settles the retained configuration, not the true optimum. Rigidity and optimality are independent, and this bears only on the former.

Evidence and sources

15 evidence entries

E-n042-wand125-mixed-68475-sqverify-fast-replay, E-n042-wand125-rect-6815-sqverify-fast-replay, E-n042-wand125-rect-68275-sqverify-fast-replay, E-wand125-rectangle-2026-10-02-source-replay, E-wand125-rectangle-2026-10-02-report, E-wand125-rectangle-2026-10-01-source-replay, E-wand125-rectangle-2026-09-28-source-replay, E-wand125-rectangle-2026-10-01-report, E-wand125-rectangle-2026-09-28-report, E-wand125-rectangle-report, E-kingbird-upper-register, E-nagamochi-lower, E-basic-grid-upper, E-karakus-strip-lower, E-nagamochi-lemma1-counterexample

[wand125 mixed bounds 2026-10-04] lower bound proof
[wand125 rectangle bounds 2026-10-02] lower bound proof
[wand125 rectangle bounds 2026-10-01] lower bound proof
[wand125 rectangle bounds 2026-09-28] lower bound proof
[wand125 rectangle bounds 2026] lower bound proof
[Kingbird] record catalogue
[Nagamochi 2005] lower bound proof
[Friedman DS7] survey
[Karakuş 2026] lower bound proof

In the frontier survey On GitHub

$s (42)$ — open

External intake, 2026-10-05. wand125’s mixed-certificate source reports $s (42) \geq 2739 / 400 = 6.8475$ (T-090), from a density of 431 rectangles of total mass $4199999 / 100000 < 42$ , accepted there at every net angle by its research copy of Tokoharu’s checker at threshold one. It is above the $2731 / 400$ this case reported, by $0.02$ . It is the source’s first mixed certificate at this count. sqverify-fast, this repository’s clean-room verifier, decided it here on 6 October 2026 at all 201 net directions, so it is also the verified lower bound: confirmed, independently re-implemented. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-02. wand125’s rectangle-density source reports the rectangle certificate rect_n42_L68275 at side $2731 / 400 = 6.8275$ , since superseded (T-090), with total mass $4199 / 100 = 41.99 < 42$ , accepted by Tokoharu’s unchanged interval checker. The complete 201-direction coverage replay here accepted it again on 3 October, 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 was the verified lower bound from its entry on 6 October (T-077) until T-090’s certificate was decided here later that day. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-10-01. wand125’s rectangle-density source reports a direct $1363 / 200 = 6.815$ certificate for this case, whose reported bound the 2026-10-02 intake above raises, with total mass $4199 / 100 = 41.99 < 42$ , 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 was also the verified lower bound until the 2026-10-02 intake above. wand125’s README says parts of the work were produced with AI assistance under human direction.

External intake, 2026-09-28. wand125’s rectangle-density source reports a direct $679 / 100 = 6.79$ certificate for this case, whose reported bound the 2026-10-01 intake above raises, with total mass $4199 / 100 = 41.99 < 42$ , 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, 2026-09-27. wand125’s rectangle-density source reports a direct $169 / 25 = 6.76$ certificate for this case, whose reported bound the 2026-09-28 intake above raises, with total mass $4199 / 100 = 41.99 < 42$ , 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.

Open. The best known packing gives $s (42) \leq 7$ , and the verified lower bound is $s (42) \geq 2739 / 400 = 6.8475$ , from wand125’s mixed rectangle-density certificate of 4 October (T-090, decided here on 2026-10-06), leaving a gap of $0.1525$ . Its rectangle-density certificate of 2 October, $2731 / 400 = 6.8275$ (T-077, replayed here on 2026-10-03 and entered on 2026-10-06), held the field earlier that day, and its certificate of 1 October, $1363 / 200 = 6.815$ (T-074), until then.

The packing

The record catalogue does not picture $n = 42$ : no arrangement has ever been found that beats the trivial $⌈ \sqrt{42} ⌉ = 7$ grid, so the grid is still the best known packing. That is a statement about what has been searched, not a proof.

The lower bound

The verified field rests on a complete decision here (E-n042-wand125-mixed-68475-sqverify-fast-replay, V3/C3) of wand125’s mixed certificate of 4 October by sqverify-fast, this repository’s clean-room measure verifier: all 201 net directions of the retained candidate on 6 October 2026, with two mutants scaled below coverage one refused. It shares no code with the source’s checker, so it decides coverage a second way with the same method; the source’s own checker was not run here. The verified lower bound was $1363 / 200$ , wand125’s rectangle-density certificate of 1 October (E-wand125-rectangle-2026-10-01-source-replay), until T-077 raised it to $2731 / 400$ , its certificate of 2 October (E-wand125-rectangle-2026-10-02-source-replay), on 6 October; this decision raised it again later that day.

Nagamochi’s general closed form, the verified lower bound before this certificate, is now a reported bound (see the correction below). Karakuş’s general bound, independently verified here and below this certificate at this $n$ , applies to every nonsquare $N \geq 8$ :

s (N) \geq \frac{1}{2} + \sqrt{N - ⌊ \sqrt{N} ⌋ + \frac{1}{4}}

Correction, 2 October 2026. Until that date the verified lower bound here was Nagamochi’s general closed form, $s (N) \geq min (⌈ \sqrt{N} ⌉, \sqrt{N - 2 ⌊ \sqrt{N} ⌋ + 1} + 1)$ , which is stronger at this $n$ and is now recorded as a reported bound. Its published proof rests on Nagamochi’s Lemma 1, which Karakuş showed false; nothing is disproved, and no packing beating it is known (review of 2 October 2026). This register had recorded that proof as verified, its own error, logged as defect D-516.

See the Frontier corpus summary for the current aggregate count; source-reported bounds are recorded separately.

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-n042-wand125-mixed-68475-sqverify-fast-replay`	replayed here	independent	`V-sqverify-fast` (first-party)
verified upper	`E-basic-grid-upper`	replayed here	independent	`V-check-basic-bounds` (first-party)