Changelog

31 October 2026 · 10 results · 8 new records · 1 change

T-252 s(44) = 7, by a closed mixed cover of points and grid-line segmentsn = 44 · V3 C3 S3 · Khalili, Gharabaghi
T-254 Couzo's new packing of 130 squares from PR #574's, 2.05e-3 below T-174n = 130 · V3 C3 S3 · Couzo · superseded
T-255 Couzo's certificates of ntumlnoob's Kaggle packings at 260 and 307, below T-225 and T-174n = 260, 307 · V3 C3 S3 · Couzo · superseded
T-256 Couzo's packings from Kaggle notebook packings at 232, 260 and 307, below T-186, T-225 and T-174n = 232, 260, 307 · V3 C3 S3 · Couzo
T-257 Daniel's new packings of 172, 204, 260 and 293 squares, 1.64e-4 to 1.18e-2 below each housen = 172, 204, 260, 293 · V3 C3 S3 · Daniel
T-258 Daniel's lower packing of 299 squares from his T-248 packing, 2.46e-3 below itn = 299 · V3 C3 S3 · Daniel
T-261 Reported grade-C local minima for 233 source configurations of field degree at most 4233 counts from 1 to 324 · V3 C1 S2 · Daniel
T-253 Polkovskyi's sub-print re-certifications of nine house packings, 1.3e-19 to 1.7e-19 below each9 counts from 154 to 271 · V0 C1 S1 · Polkovskyi
T-259 Polkovskyi's re-solve of Couzo's packing of 130 squares, 2.08e-12 below T-254n = 130 · V3 C3 S1 · Polkovskyi
T-260 Polkovskyi's sub-print re-certifications at 105, 146, 156, 229 and 308, 1e-41 to 1e-21 below eachn = 105, 146, 156, 229, 308 · V0 C1 S1 · Polkovskyi
T-256 is now the record packing at 232 and 307
T-257 is now the record packing at 172, 204, 260 and 293
T-258 is now the record packing at 299
T-259 is now the record packing at 130
Daniel’s, Couzo’s and Polkovskyi’s Certificates Become the House Best Packings at Eight CountsS3 · Issue #824 · Issue #843 · Issue #844 · Issue #845 · Issue #846 · PR #852

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26 October 2026 · 5 results · 1 new record · 1 bound beyond 324 · 2 changes

T-248 Daniel's new arrangement at 299, 1.92e-2 below the catalogue's constructionn = 299 · V3 C3 S3 · Daniel · superseded
T-250 Hosono reports a Lean kernel proof of Daniel's Valid7, and so of s(k² − 3) = k for every k ≥ 613 counts from 33 to 321 · V0 C0 S3 · Hosono
T-251 Hosono reports a Lean kernel proof of Daniel's ValidTilt9, so s(k² − 4) = k for every k ≥ 811 counts from 60 to 320 · V0 C0 S3 · Hosono
T-247 Kim's exact degree-21 side at 297, as reported, of the packing the record creditsn = 297 · V3 C3 S2 · Kim
T-249 Daniel reports a Lean kernel proof that s(12) ≥ 15680/3951, the bound T-049 holdsn = 12 · V0 C0 S2 · Daniel
T-242 is now the record packing at 268
1 upper bound beyond n = 324n = 344 · Amund Kristiansen
A Concise Changelog, and News for the StoriesS2 · PR #840
The Project Changelog Gets a Page, Under a Project MenuS2 · PR #837 · The site design record

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25 October 2026 · 4 results · 3 new records · 147 bounds beyond 324 · 3 changes

T-245 Jatloe reports a Lean proof that the 4 × 3 grid is optimal, s(12) = 4n = 12 · V0 C0 S5 · jatloe, Gurevitch, ctjlewis, Ahmed
T-243 Daniel's new arrangement at 182, 3.17e-3 below T-174n = 182 · V3 C3 S3 · Daniel
T-246 Reported degree-7213 minimal polynomial of the s(29) record siden = 29 · V0 C0 S3 · Daniel
T-244 Kim's exact algebraic sides at 172 and 263, as reported, of packings the record creditsn = 172, 263 · V3 C3 S2 · Kim
T-240 is now the record packing at 303
T-242 is now the record packing at 304
T-243 is now the record packing at 182
122 upper bounds beyond n = 324n = 330–997 · Francisco Couzo · 6 since superseded
13 upper bounds beyond n = 324n = 365–970 · Francisco Couzo · 7 since superseded
12 upper bounds beyond n = 324n = 334–504 · Amund Kristiansen
Couzo’s Packings Become the House Best Packings at 268 and 303S3 · Issue #763 · PR #794
Daniel’s New Arrangement Becomes the House Best Packing at 182S3 · Issue #806 · PR #814
Jatloe Reports a Lean Proof for 12 SquaresS5 · Issue #810 · PR #812

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24 October 2026 · 12 results · 9 new records · 517 bounds beyond 324 · 2 changes

T-232 Mixed rectangle-measure lower bound s(31) ≥ 2391/400 verified, on a 2,073-direction declared netn = 31 · V3 C3 S3 · Hosono
T-233 Mixed rectangle-measure lower bounds reported at n = 19, 20, 26, 27, on a 2,073-direction netn = 19, 20, 26, 27 · V0 C1 S3 · Hosono
T-236 Daniel reports Lean proofs of Fang's Theorems 1 and 2, s(n² − n + 1) < n for n ≥ 15n = 211, 241, 273, 307 · V0 C0 S3 · Daniel
T-237 s(2), s(3), s(4) = 2 and s(5) = 2 + 1/√2 = 2.70710678…, the classical first casesn = 2–5 · V3 C3 S3 · Göbel; Friedman
T-240 Couzo's 268 and 303 of #763, below T-174 and T-239, from Kaggle notebook packingsn = 268, 303 · V3 C3 S3 · Couzo
T-241 Couzo's certificate at 304 of Tymchishin's Kaggle notebook packing, 2.53e-3 below T-186n = 304 · V3 C3 S3 · Couzo · superseded
T-231 Polkovskyi's lower re-solve at 204 of Couzo's packing, 1.48e-11 below T-229n = 204 · V3 C3 S2 · Polkovskyi · superseded
T-234 Hosono reports a hypothesis-free Lean proof that s(21) = 5, by the point-only routen = 21 · V0 C0 S2 · Hosono
T-235 Hosono reports a Lean kernel proof that s(59) = 8, on Daniel's mixed-measure checkern = 59 · V0 C0 S2 · Hosono
T-238 chelokot reports Lean proofs of s(2), s(3), s(4) = 2, s(5) = 2.707…; Daniel, rectangle theoremsn = 2–5 · V0 C0 S2 · Vlasenko; Daniel
T-239 Kristiansen's smaller n303 packing, found from Daniel's packing, certified exactlyn = 303 · V3 C3 S2 · Kristiansen · superseded
T-242 Daniel's exact descents of public Kaggle packings at 268 and 304, below the reports before themn = 268, 304 · V3 C3 S2 · Daniel
T-174 is now the record packing at 70, 153 and 207
T-193 is now the record packing at 102, 123, 258, 259, 295 and 302
184 upper bounds beyond n = 324n = 333–997 · Francisco Couzo · 68 since superseded
138 upper bounds beyond n = 324n = 328–989 · Mykhailo Polkovskyi · 131 since superseded
42 upper bounds beyond n = 324n = 327–975 · Kirill Tymchishin · 15 since superseded
37 upper bounds beyond n = 324n = 377–994 · Francisco Couzo · all since superseded
35 upper bounds beyond n = 324n = 377–984 · Francisco Couzo · all since superseded
18 upper bounds beyond n = 324n = 377–981 · kyleluecke · 11 since superseded
15 upper bounds beyond n = 324n = 376–984 · Kirill Tymchishin · 5 since superseded
11 upper bounds beyond n = 324n = 338–875 · alexchilton · 7 since superseded
11 upper bounds beyond n = 324n = 330–816 · Vladislav Lavrentev · all since superseded
10 upper bounds beyond n = 324n = 335–998 · Francisco Couzo · 2 since superseded
9 upper bounds beyond n = 324n = 364–994 · aqibrazadev · 8 since superseded
7 upper bounds beyond n = 324n = 653–923 · Vladislav Lavrentev · 4 since superseded
Daniel Reports Lean Proofs of Fang’s Theorems, and the Register Takes In $s (2)$ to $s (5)$ S3 · Issue #784 · Issue #785 · PR #750
Hosono Raises the Lower Bound for 31 Squares to 5.9775S3 · Issue #446 · Issue #534 · PR #745

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23 October 2026 · 1 result · 1 new record · 396 bounds beyond 324 · 3 changes

T-230 Daniel's smaller n303 packing, started from Ostashev's packing, certified exactlyn = 303 · V3 C3 S3 · Daniel · superseded
T-229 is now the record packing at 105
253 upper bounds beyond n = 324n = 325–994 · Francisco Couzo · 214 since superseded
77 upper bounds beyond n = 324n = 372–995 · Mykhailo Polkovskyi · all since superseded
21 upper bounds beyond n = 324n = 547–997 · Francisco Couzo · all since superseded
20 upper bounds beyond n = 324n = 409–997 · Francisco Couzo · all since superseded
12 upper bounds beyond n = 324n = 332–379 · Francisco Couzo · 6 since superseded
4 upper bounds beyond n = 324n = 340–380 · Francisco Couzo · all since superseded
2 upper bounds beyond n = 324n = 338–340 · Francisco Couzo · all since superseded
2 upper bounds beyond n = 324n = 344–378 · Francisco Couzo · all since superseded
2 upper bounds beyond n = 324n = 703–994 · Kevin Fang · all since superseded
1 upper bound beyond n = 324n = 378 · Francisco Couzo · all since superseded
1 upper bound beyond n = 324n = 344 · Francisco Couzo · all since superseded
1 upper bound beyond n = 324n = 343 · Evan Daniel · all since superseded
Couzo’s and Polkovskyi’s Certificates at Four Counts ConfirmedS3 · Issue #670 · Issue #672 · Issue #675 · Issue #719 · PR #725
Daniel’s Smaller Packing of 303 Squares AdoptedS3 · Issue #728 · PR #731
The Visual Atlas Follows the ScrollS2 · PR #746 · The site design record

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22 October 2026 · 5 results · 2 new records · 425 bounds beyond 324 · 1 change

T-226 Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678n = 204 · V3 C3 S3 · Couzo · superseded
T-227 Couzo's refinements of de Winter's 227 and 229, below T-196 at bothn = 227, 229 · V3 C3 S3 · Couzo · superseded
T-225 Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported theren = 204, 227, 229, 260 · V3 C3 S2 · Daniel
T-228 Couzo's 204 of #675, below his #670, since superseded by his #678n = 204 · V3 C3 S2 · Couzo · superseded
T-229 Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225n = 105, 204 · V3 C3 S2 · Polkovskyi
T-225 is now the record packing at 227 and 229
111 upper bounds beyond n = 324n = 330–997 · Mykhailo Polkovskyi · 109 since superseded
109 upper bounds beyond n = 324n = 328–996 · Francisco Couzo · all since superseded
78 upper bounds beyond n = 324n = 419–998 · Francisco Couzo · all since superseded
72 upper bounds beyond n = 324n = 330–993 · Francisco Couzo · all since superseded
44 upper bounds beyond n = 324n = 330–996 · Francisco Couzo · all since superseded
11 upper bounds beyond n = 324n = 464–998 · Joost de Winter · all since superseded
Contributor Profiles With Records, Results and CreditsS3 · PR #578 · Contributor records · Attribution rules

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21 October 2026 · 2 results · 1 new record · 184 bounds beyond 324 · 3 changes

T-223 Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 boundsn = 204, 260 · V3 C3 S3 · Couzo · superseded
T-224 Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactlyn = 270 · V3 C3 S3 · Daniel
T-224 is now the record packing at 270
100 upper bounds beyond n = 324n = 328–998 · Darius Muglich · 97 since superseded
83 upper bounds beyond n = 324n = 330–998 · Francisco Couzo · all since superseded
1 upper bound beyond n = 324n = 343 · Evan Daniel · all since superseded
Couzo’s Smaller Packings of 204 and 260 Squares AdoptedS3 · Issue #678 · PR #706
Daniel’s Smaller Packing of 270 Squares AdoptedS3 · Issue #703 · PR #708
De Winter’s Smaller Packings of 227 and 260 Squares AdoptedS3 · Issue #668 · PR #704

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20 October 2026 · 5 results · 6 new records · 4 changes

T-218 Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactlyn = 236 · V3 C3 S3 · Daniel
T-219 Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactlyn = 271 · V3 C3 S3 · Daniel
T-220 Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packingn = 183 · V3 C3 S3 · Daniel
T-221 Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactlyn = 308 · V3 C3 S3 · Daniel
T-222 Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactlyn = 154 · V3 C3 S3 · Daniel
T-197 is now the record packing at 155
T-218 is now the record packing at 236
T-219 is now the record packing at 271
T-220 is now the record packing at 183
T-221 is now the record packing at 308
T-222 is now the record packing at 154
Daniel’s Smaller Packings at 155, 229, 236 and 271 Squares AdoptedS3 · Issue #659 · Issue #664 · Issue #669 · Issue #674 · PR #676 · PR #692
Daniel’s Packing of 183 Squares Below Side 14S3 · Issue #680 · PR #687
Daniel’s Smaller Packings of 154 and 308 Squares AdoptedS3 · Issue #689 · Issue #694 · PR #699
Hosono Reports Stronger Lower Bounds at 18 Counts, Four VerifiedS3 · Issue #446 · PR #671 · PR #688

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19 October 2026 · 23 results · 29 new records · 1 change

T-196 de Winter reports three smaller rational upper bounds at 227, 229 and 260n = 227, 229, 260 · V3 C3 S3 · de Winter · superseded
T-197 Daniel's smaller n155 packing, started from Ostashev's packingn = 155 · V3 C3 S3 · Daniel
T-200 Mixed rectangle-measure lower bound s(38) ≥ 2631/400 reported, on a 2,073-direction declared netn = 38 · V0 C0 S3 · Hosono
T-201 Mixed rectangle-measure lower bound s(40) ≥ 1347/200 reported, on a 2,073-direction declared netn = 40 · V0 C0 S3 · Hosono
T-202 Mixed rectangle-measure lower bound s(41) ≥ 2721/400 reported, on a 2,073-direction declared netn = 41 · V0 C0 S3 · Hosono
T-203 Mixed rectangle-measure lower bound s(43) ≥ 2777/400 reported, on a 2,073-direction declared netn = 43 · V0 C0 S3 · Hosono
T-204 Mixed rectangle-measure lower bound s(51) ≥ 3011/400 reported, on a 2,073-direction declared netn = 51 · V0 C0 S3 · Hosono
T-205 Mixed rectangle-measure lower bound s(68) ≥ 853/100 verified, on the 201-direction netn = 68 · V3 C3 S3 · Hosono
T-206 Mixed rectangle-measure lower bound s(84) ≥ 3769/400 verified, on the 201-direction netn = 84 · V3 C3 S3 · Hosono
T-207 Mixed rectangle-measure lower bound s(85) ≥ 379/40 verified, on the 201-direction netn = 85 · V3 C3 S3 · Hosono
T-208 Mixed rectangle-measure lower bound s(86) ≥ 3811/400 reported, on the 201-direction netn = 86 · V0 C0 S3 · Hosono
T-209 Mixed rectangle-measure lower bound s(88) ≥ 3853/400 reported, on the 201-direction netn = 88 · V0 C0 S3 · Hosono
T-210 Mixed rectangle-measure lower bound s(89) ≥ 1937/200 reported, on the 201-direction netn = 89 · V0 C0 S3 · Hosono
T-211 Mixed rectangle-measure lower bound s(91) ≥ 783/80 reported, on the 201-direction netn = 91 · V0 C0 S3 · Hosono
T-212 Mixed rectangle-measure lower bound s(90) ≥ 489/50 reported, on the 201-direction netn = 90 · V0 C0 S3 · Hosono
T-213 Mixed rectangle-measure lower bound s(92) ≥ 493/50 reported, on the 201-direction netn = 92 · V0 C0 S3 · Hosono
T-214 Mixed rectangle-measure lower bound s(106) ≥ 10503/1000 reported, on the 201-direction netn = 106 · V0 C0 S3 · Hosono
T-215 Mixed rectangle-measure lower bound s(107) ≥ 529/50 reported, on the 201-direction netn = 107 · V0 C0 S3 · Hosono
T-216 Mixed rectangle-measure lower bound s(108) ≥ 531/50 verified, on the 201-direction netn = 108 · V3 C3 S3 · Hosono
T-217 Mixed rectangle-measure lower bound s(110) ≥ 1073/100 reported, on the 201-direction netn = 110 · V0 C0 S3 · Hosono
T-198 Daniel's opening n229 dilation certificate, after Couzo and de Wintern = 229 · V3 C3 S2 · Daniel · superseded
T-199 Daniel's smaller n229 update, replacing the opening certificaten = 229 · V3 C3 S2 · Daniel · superseded
T-195 A tighter rational certificate of Schadt and Ellsworth's 29-square packing, 1.57e-36 below T-155n = 29 · V3 C3 S1 · PingYou Ltd
T-174 is now the record packing at 68, 84, 86, 103, 108, 110, 127, 131, 132, 179, 180, 181, 206, 208, 209, 210, 228, 238, 239, 240, 261, 263, 269, 272, 273, 297, 305 and 306
T-195 is now the record packing at 29
Fitzharding’s Reported 17-Square Proof Admitted on Its Own BuildS3 · Issue #565 · PR #645

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17 October 2026 · 8 new records · 1 change

T-186 is now the record packing at 129, 156, 175, 176 and 264
T-187 is now the record packing at 146
T-188 is now the record packing at 211
T-190 is now the record packing at 237
Smaller Confirmed Packings Adopted at 15 Square CountsS3 · Issue #506 · Issue #593 · Issue #606 · Issue #607 · PR #657

16 October 2026 · 7 results · 267 bounds beyond 324 · 4 changes

T-194 Fitzharding reports a Lean proof that Wainwright's packing is optimal, s(19) = 3 + 4√2/3n = 19 · V0 C1 S5 · Fitzharding
T-189 Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon boundsn = 237 · V3 C3 S3 · Couzo · superseded
T-191 s(95) = 10, by a mixed cover of points and grid-line segmentsn = 95, 96 · V3 C3 S3 · Greff
T-192 Continuous-pose lower bound s(28) > 1149/200 reported, by two of the source's programsn = 28 · V0 C1 S3 · Hosono
T-188 A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangementn = 211 · V3 C3 S2 · Couzo
T-190 A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificaten = 237 · V3 C3 S2 · Couzo
T-193 Exact algebraic sides at sixteen counts from Daniel, of packings the record credits16 counts from 88 to 302 · V3 C3 S2 · Daniel
252 upper bounds beyond n = 324n = 339–991 · Francisco Couzo · all since superseded
15 upper bounds beyond n = 324n = 339–804 · Francisco Couzo · all since superseded
Fitzharding Reports a Lean Proof for 19 SquaresS5 · Issue #640 · PR #651
Greff Reports an Optimum for 95 SquaresS3 · Issue #610 · PR #627
Hosono Reports a Stronger Lower Bound for 28 SquaresS3 · PR #627
Regularized Drawings for Six Re-Housed Counts, Including 176S3 · PR #644

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15 October 2026 · 3 results · 1 new record · 4 changes

T-185 A tighter packing of 175 squares from Couzo, superseded by Ostashev's compositionn = 175 · V3 C3 S2 · Couzo · superseded
T-187 A smaller packing of 146 squares from Daniel's chain search, started from Xu's packingn = 146 · V3 C3 S2 · Daniel
T-186 Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there12 counts from 129 to 304 · V3 C3 S1 · Polkovskyi
T-182 is now the record packing at 241
Atlas Posters Include the October Packing ResultsS2 · PR #619 · Current 1–100 packing infographic · Current 1–324 packing poster · Lower-bounds paper · Publication evidence · Atlas and downloadable editions · Published release and verification · Site v0.7.7; Infographics v0.6.1; New Lower Bounds for Square Packing for n = 11 v0.4.7
Every Selected Validation Check Now RunsS3 · PR #620 · Validation tiers
Smaller Verified Packings for 18 Square CountsS3 · Issue #503 · Issue #504 · Issue #561 · Issue #562 · Issue #564 · Issue #566 · Issue #567 · Issue #568 · Issue #569 · Issue #570 · Issue #573 · Issue #574 · Issue #575 · Issue #583 · Issue #586 · Issue #593 · PR #614
Significance Levels for News and Pull RequestsS3 · PR #621 · The significance rubric · Maintaining the project changelog

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14 October 2026 · 30 results · 39 bounds beyond 324 · 9 changes

T-163 s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fangn = 211, 241, 273, 307 · V3 C1 S4 · Fang · superseded
T-180 Closed-form lower bound s(N) ≥ a(N) for every non-square N ≥ 49, reported264 counts from 50 to 323 · V0 C1 S4 · Hosono
T-157 s(40) > 3359/500 = 6.718, by T-145's continuous-pose kernel on a density with 163 new supportsn = 40 · V3 C3 S3 · Guzhou0806
T-158 New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactlyn = 175, 232, 264 · V3 C3 S3 · Ostashev · superseded
T-160 A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactlyn = 154 · V3 C3 S3 · Daniel · superseded
T-161 Exact rational certificates at 156, 176, 237 and 271 from Couzon = 156, 176, 237, 271 · V3 C3 S3 · Couzo · superseded
T-162 New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactlyn = 129, 204, 303 · V3 C3 S3 · Ostashev · superseded
T-164 A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banksn = 302 · V3 C3 S3 · PingYou Ltd · superseded
T-172 A packing of 154 squares from Couzo, continuing Daniel's arrangementn = 154 · V3 C3 S3 · Couzo · superseded
T-175 New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactlyn = 211, 241, 304 · V3 C3 S3 · Ostashev · superseded
T-176 Mixed rectangle-measure lower bound s(39) ≥ 533/80 reported, on a 2,073-direction declared netn = 39 · V0 C1 S3 · Hosono
T-177 Mixed rectangle-measure lower bound s(44) ≥ 2799/400 reported, on a 2,073-direction declared netn = 44 · V0 C1 S3 · Hosono · superseded
T-178 Mixed rectangle-measure lower bound s(37) ≥ 2599/400 verified, on a 2,073-direction declared netn = 37 · V3 C3 S3 · Hosono
T-179 Linear-measure lower bound s(n) ≥ 306/25 = 12.24 for n = 145…149, reportedn = 145–149 · V0 C1 S3 · Hosono
T-181 Exact rational certificates at 211, 241 and 302 from Couzon = 211, 241, 302 · V3 C3 S3 · Couzo · superseded
T-182 Exact rational certificates at 154, 211 and 241 from Couzon = 154, 211, 241 · V3 C3 S3 · Couzo
T-183 Polkovskyi's certificate for 237 squares, the smallest known side there when registeredn = 237 · V3 C3 S3 · Polkovskyi · superseded
T-184 s(76) = 9, by a mixed cover of points and grid-line segmentsn = 76 · V3 C3 S3 · Greff
T-159 Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layoutn = 102 · V3 C3 S2 · Calle · superseded
T-165 A tighter certificate at 176 from Couzo, published with no issuen = 176 · V3 C3 S2 · Couzo · superseded
T-167 A tighter rational realization of PingYou Ltd's packing of 207 squaresn = 207 · V3 C3 S2 · PingYou Ltd · superseded
T-168 A further exact refinement of Xu's packing of 146 squares, after Barnett'sn = 146 · V3 C3 S2 · PingYou Ltd · superseded
T-169 A further exact refinement of Xu's packing of 261 squares, after Barnett'sn = 261 · V3 C3 S2 · PingYou Ltd · superseded
T-173 Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements theren = 153, 207 · V3 C3 S2 · Polkovskyi · superseded
T-155 Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace59 counts from 28 to 300 · V3 C3 S1 · Levy
T-156 s(290) ≤ 17.5355339059327376…, by monotonicity from T-155, with its own exact certificaten = 290 · V3 C3 S1 · Levy
T-166 A tighter rational realization of PingYou Ltd's packing of 153 squaresn = 153 · V3 C3 S1 · PingYou Ltd · superseded
T-170 An exact contact refinement of Xu's packing of 105 squares, after Couzo'sn = 105 · V3 C3 S1 · PingYou Ltd · superseded
T-171 An exact contact refinement of SQUISH's packing of 272 squaresn = 272 · V3 C3 S1 · PingYou Ltd · superseded
T-174 Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own68 counts from 68 to 308 · V3 C3 S1 · PingYou Ltd
14 upper bounds beyond n = 324n = 380–814 · Kevin Fang · all since superseded
9 upper bounds beyond n = 324n = 339–382 · Francisco Couzo · 8 since superseded
5 upper bounds beyond n = 324n = 338–378 · Francisco Couzo · 3 since superseded
4 upper bounds beyond n = 324n = 339–382 · Francisco Couzo · all since superseded
3 upper bounds beyond n = 324n = 339–375 · Evan Daniel · all since superseded
3 upper bounds beyond n = 324n = 365–460 · Evan Daniel · all since superseded
1 upper bound beyond n = 324n = 932 · Kevin Fang · all since superseded
Basic and Advanced Tutorials Under ExplanationsS4 · PR #604
Initial Reviews of the 17-Square Proof ReportsS3 · Issue #557 · Issue #565 · PR #592 · PR #595
Audit Covers Every Open Upper-Bound ReportS3 · PR #601
Fang and Hosono Report New Bound FamiliesS4 · Issue #446 · Issue #503 · Issue #504 · Issue #561 · Issue #562 · Issue #564 · Issue #566 · Issue #567 · Issue #568 · Issue #569 · Issue #570 · Issue #573 · Issue #574 · Issue #575 · Issue #583 · Issue #586 · Issue #593 · Issue #594 · Issue #600 · Issue #603 · PR #602
A Stronger 40-Square Bound and Four Smaller PackingsS3 · Issue #485 · Issue #528 · Issue #529 · Issue #552 · PR #598
Release Histories for the Site, Papers and PostersS4 · PR #590 · Current About changelog · Current 1–100 packing infographic · Current 1–324 packing poster · Current lower-bounds explainer · Publication evidence · GitHub changelog · Component release rules · Site v0.6.0; Infographics v0.6.0; New Lower Bounds for Square Packing for n = 11 v0.4.6
Regularized Packings With Originals for ComparisonS4 · PR #511 · PR #530 · PR #548 · PR #549 · PR #559 · PR #580 · PR #585 · PR #587 · PR #591
Seventeen Close Packing Comparisons Resolved ExactlyS2 · PR #605 · Publication evidence · The intake runbook · The decided comparisons · Site v0.6.3
Result Ratings and Verification Evidence AuditedS3 · Issue #238 · Issue #256 · PR #563

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13 October 2026 · 2 results · 5 changes

T-153 Candidate proof that Bidwell's 1998 packing is optimal, s(17) = 4.67553009360455…n = 17 · V0 C1 S5 · Overbeck
T-154 Fitzharding reports a Lean proof that s(17) = 4.67553009360455…n = 17 · V3 C1 S5 · Fitzharding
Verification Gaps in 92 Packing RecordsS3 · PR #581
Fitzharding Reports a Lean Proof for 17 SquaresS5 · Issue #565 · PR #588
Overbeck Reports a Candidate Proof for 17 SquaresS5 · Issue #557 · PR #560
A Second Review of Overbeck’s 17-Square Candidate ProofS3 · Issue #557 · PR #577
Rules for Verifying Regularized DrawingsS2 · PR #497 · PR #509 · Regularization policy · Assessment · Retained regularizer

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12 October 2026 · 6 results · 3 new records · 9 bounds beyond 324 · 5 changes

T-147 A packing of 207 squares from two placed patches of Xu's packing of 70n = 207 · V3 C3 S3 · PingYou Ltd · superseded
T-148 A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new anglen = 153 · V3 C3 S3 · PingYou Ltd · superseded
T-149 New packings of 155, 156 and 211 squares, each with an exact rational certificaten = 155, 156, 211 · V3 C3 S3 · Ostashev · superseded
T-151 Nine exact rational packings from SQUISH's fourth requestn = 176, 181, 238, 239, 270–272, 304, 305 · V3 C3 S3 · Chaoweeraprasit · superseded
T-152 Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squaresn = 304, 305 · V3 C3 S3 · Couzo · superseded
T-150 Exact rational refinements of Xu's packings of 146 and 261 squaresn = 146, 261 · V3 C3 S2 · Barnett · superseded
T-134 is now the record packing at 267
T-139 is now the record packing at 126
T-146 is now the record packing at 292
9 upper bounds beyond n = 324n = 338–381 · Francisco Couzo · 7 since superseded
Corrections to Result Reports Stay VisibleS2 · PR #531 · Intake workflow
Smaller Certified Packings for 42 Square CountsS3 · Issue #501 · Issue #503 · Issue #504 · Issue #505 · PR #512 · PR #516 · PR #518
PingYou Names PY as Its AI AssistantS2 · Issue #503 · Issue #504 · PR #558 · Current 207-square PingYou record · Current 153-square PingYou record · Publication evidence
Publication Checks Follow Code DependenciesS2 · PR #507 · Publication and validation workflow
Nine Smaller Packings From Chaoweeraprasit and CouzoS3 · Issue #500 · Issue #506 · PR #553

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11 October 2026 · 15 results · 4 bounds beyond 324 · 6 changes

T-132 Mixed rectangle-measure lower bound s(29) ≥ 291/50 reported, on a 2,073-direction declared netn = 29 · V0 C0 S3 · Hosono
T-133 s(40) > 335427/50000 = 6.70854, by a clipped-corner transfer on a 401-direction netn = 40 · V3 C3 S3 · Guzhou0806 · superseded
T-134 Six exact rational certificates from Couzon = 132, 237, 263, 267, 270, 303 · V3 C3 S3 · Couzo
T-135 A packing of 308 squares below the grid, from Fangn = 308 · V3 C3 S3 · Fang · superseded
T-136 Seventeen exact rational packings from SQUISH's third request17 counts from 131 to 307 · V3 C3 S3 · Chaoweeraprasit · superseded
T-138 Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered16 counts from 84 to 306 · V3 C3 S3 · Polkovskyi · superseded
T-140 Six more exact rational certificates from Couzon = 132, 175, 209, 237, 270, 305 · V3 C3 S3 · Couzo · superseded
T-141 Exact rational certificates at 132 and 308 from Daniel's third record huntn = 132, 308 · V3 C3 S3 · Daniel · superseded
T-142 Mixed rectangle-measure lower bound s(28) ≥ 2297/400 reported, on a 2,073-direction declared netn = 28 · V0 C0 S3 · Hosono
T-143 Mixed rectangle-measure lower bound s(30) ≥ 2357/400 reported, on a 2,073-direction declared netn = 30 · V0 C0 S3 · Hosono
T-144 Linear-measure lower bound s(n) ≥ 563/50 = 11.26 for n = 122…126, reportedn = 122–126 · V0 C0 S3 · Hosono
T-145 s(40) > 1340000/199529 = 6.7158157…, by a continuous-pose interval kerneln = 40 · V3 C3 S3 · Guzhou0806 · superseded
T-137 An exact rational refinement of Xu's packing of 70 squaresn = 70 · V3 C3 S2 · Deleeuw · superseded
T-139 A more precise certificate of Xu's 126-square arrangement, from Danieln = 126 · V3 C3 S2 · Daniel
T-146 Exact rational certificates at fifteen counts from Daniel's regularized record lists15 counts from 70 to 303 · V3 C3 S2 · Daniel
2 upper bounds beyond n = 324n = 343–344 · Evan Daniel · all since superseded
2 upper bounds beyond n = 324n = 343–344 · Kevin Fang · all since superseded
Publication Audits Show Progress and FailuresS2 · PR #496 · Publication and validation workflow
Fuller Credit for Daniel’s ResultsS2 · Issue #256 · Issue #296 · Issue #316 · PR #508 · Current contributor and result overview · Publication evidence
Exact Certificates and a Stronger 40-Square BoundS3 · Issue #375 · Issue #446 · Issue #451 · Issue #465 · Issue #470 · Issue #476 · Issue #481 · Issue #483 · Issue #484 · Issue #485 · Issue #488 · Issue #489 · PR #487
Narrower Text Columns and Responsive NavigationS2 · PR #513 · Current project overview · Current About page · Publication evidence · Front-door prose review · Reader-document prose review
Shared Navigation Connects the Atlas, Results and PapersS4 · PR #462 · Overview · Atlas · Results · Papers and downloads · Publication evidence
Share a Result With the Squares ProjectS2 · PR #502 · Project homepage · Reporting results on About · Publication evidence

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10 October 2026 · 1 result · 2 bounds beyond 324 · 5 changes

T-131 s(132) ≤ 11.987099332245063… from Daniel's record huntn = 132 · V3 C3 S3 · Daniel · superseded
2 upper bounds beyond n = 324n = 375–378 · Francisco Couzo · 1 since superseded
Daniel’s Exact Certificates for 132 and 155 SquaresS3 · Issue #451 · Issue #465 · PR #478
Couzo Reports Smaller Packings at 375 and 378 SquaresS3 · PR #479
Five Illustrations of Packing Search and CertificationS3 · PR #477
Consistent Typography Across the PapersS2 · PR #492 · Current Part I paper · Current Part I PDF · Current Part II paper · Current Part II PDF · Current Part III paper · Current Part III PDF · Current methods tutorial · Current methods tutorial PDF · Publication evidence · New Lower Bounds for Square Packing for n = 11 v0.4.5; How Record Square Packings Are Found v0.2.1
Compare Packings on Screen and on PaperS4 · PR #474

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9 October 2026 · 14 results · 20 bounds beyond 324 · 10 changes

T-119 Three exact feasible upper bounds at n=266,270,272 from new source arrangementsn = 266, 270, 272 · V3 C3 S3 · Daniel
T-125 Complete rational construction reports at 25 counts25 counts from 51 to 295 · V3 C3 S3 · Xu · superseded
T-126 Undilated n51 construction over Q(sqrt2)n = 51 · V3 C3 S3 · Xu
T-127 Fourteen rational refinements, seventeen complete source cases17 counts from 88 to 239 · V3 C3 S3 · Gupta
T-128 Eight complete rational refinements from Couzon = 105, 108, 127, 131, 155, 180, 228, 306 · V3 C3 S3 · Couzo · superseded
T-130 Five follow-up rational refinements from Couzon = 84, 86, 105, 175, 270 · V3 C3 S3 · Couzo · superseded
T-117 Exact rational ceiling refinements at n = 105, 292n = 105, 292 · V3 C3 S2 · Rehwaldt · superseded
T-118 Exact rational ceiling refinements at n = 68n = 68 · V3 C3 S2 · Rehwaldt · superseded
T-120 Reported s(102) ≤ alpha_102 with a degree-8 exact side formn = 102 · V0 C0 S2 · Daniel · superseded
T-121 Reported s(106) ≤ alpha_106 with a degree-32 exact side formn = 106 · V0 C0 S2 · Daniel
T-122 Reported s(152) ≤ alpha_152 with a degree-40 exact side formn = 152 · V0 C0 S2 · Daniel
T-123 Reported s(177) ≤ alpha_177 with a degree-32 exact side formn = 177 · V0 C0 S2 · Daniel
T-124 Reported non-strict local minima for 178 source configurations178 counts from 1 to 324 · V0 C0 S2 · Daniel
T-129 Three dated Daniel certificate reports with historical source custodyn = 105, 130, 292 · V0 C0 S1 · Daniel · superseded
20 upper bounds beyond n = 324n = 327–379 · Francisco Couzo · 17 since superseded
Couzo Reports Eight Smaller Exact PackingsS3 · Issue #451 · Issue #465 · PR #460
Couzo Reports Five Further Exact PackingsS3 · Issue #451 · PR #469
Couzo Reports Packings Beyond 324 SquaresS3 · PR #466
Daniel’s Smaller Packings at 266, 270 and 272 SquaresS3 · Issue #399 · Issue #419 · Issue #420 · PR #441
Gupta’s Refinements Improve 14 Packing BoundsS3 · Issue #438 · PR #448 · PR #459
Hosono Reports Seven Stronger Lower BoundsS3 · Issue #446 · PR #450
Hosono’s Twelve Certificates Narrow the 17-Square SearchS3 · Issue #472 · PR #475
A Survey of How Square Packings Are FoundS4 · PR #458
Ryu Reports Polynomial Deficiency BoundsS4 · Issue #414 · PR #439 · PR #440
Xu’s Verified Packings Include an Exact 51-Square ConstructionS3 · Issue #432 · PR #442

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8 October 2026 · 1 result · 1 change

T-116 s(n) ≤ S_n at nine counts, from the second SQUISH update’s new rational packings9 counts from 88 to 302 · V3 C3 S3 · Chaoweeraprasit · superseded
Readable Formulas in the Frontier TablesS2 · PR #456 · Frontier table · Static rendering and deployment work

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7 October 2026 · 3 results

T-113 s(n) ≤ S_n at ten counts, n = 108 to 303, from new SQUISH rational packings10 counts from 108 to 303 · V3 C3 S3 · Chaoweeraprasit · superseded
T-114 s(153) ≤ 7250614903299225/562949953421312, from a SQUISH rational packingn = 153 · V3 C3 S3 · Chaoweeraprasit · superseded
T-115 s(n) ≤ S_n at twelve counts, from the SQUISH update’s new and smaller rational packings12 counts from 123 to 263 · V3 C3 S3 · Chaoweeraprasit · superseded

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6 October 2026 · 18 results

T-095 s(12) ≥ 7943/2000 = 3.9715, Daniel's points dilated and re-weightedn = 12 · V3 C3 S3 · Ryu
T-096 Mixed rectangle-measure lower bound replayed at n = 18, on a net the certificate declaresn = 18 · V3 C3 S3 · Hosono · superseded
T-097 Mixed rectangle-measure lower bound verified at n = 66n = 66 · V3 C3 S3 · Hosono
T-099 Mixed rectangle-measure lower bound verified at n = 18, on a finer declared netn = 18 · V3 C3 S3 · Hosono · superseded
T-100 Mixed rectangle-measure lower bound verified at n = 19, past the best 18-square packingn = 19 · V3 C3 S3 · Hosono · superseded
T-102 Mixed rectangle-measure lower bound verified at n = 18, on the finest declared net yetn = 18 · V3 C3 S3 · Hosono
T-103 Mixed rectangle-measure lower bound verified at n = 19, on the finest declared net yetn = 19 · V3 C3 S3 · Hosono
T-104 Mixed rectangle-measure lower bound verified at n = 20, on a declared netn = 20 · V3 C3 S3 · Hosono
T-105 Mixed rectangle-measure lower bound verified at n = 26, on a declared netn = 26 · V3 C3 S3 · Hosono
T-106 Mixed rectangle-measure lower bound verified at n = 27, on a declared netn = 27 · V3 C3 S3 · Hosono
T-107 Mixed rectangle-measure lower bound verified at n = 28, on a declared netn = 28 · V3 C3 S3 · Hosono
T-108 Mixed rectangle-measure lower bound verified at n = 29, on a declared netn = 29 · V3 C3 S3 · Hosono
T-109 Mixed rectangle-measure lower bound verified at n = 30, on a declared netn = 30 · V3 C3 S3 · Hosono
T-110 Mixed rectangle-measure lower bound verified at n = 39, on a declared netn = 39 · V3 C3 S3 · Hosono
T-111 Mixed rectangle-measure lower bound verified at n = 41, on a declared netn = 41 · V3 C3 S3 · Hosono
T-112 Trump's packing is the only optimal packing of eleven squares, up to symmetryn = 11 · V3 C2 S3 · Levy
T-098 Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side48 counts from 68 to 307 · V3 C3 S2 · Daniel
T-101 Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side77 counts from 28 to 300 · V3 C3 S2 · Daniel

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5 October 2026 · 7 results

T-090 Mixed rectangle-measure lower bounds verified at 17 counts in n = 42…9617 counts from 42 to 96 · V3 C3 S3 · Hosono
T-091 Mixed rectangle-measure lower bounds verified at 17 counts in n = 53…95n = 53, 54, 58, 70–73, 76, 87–95 · V3 C3 S3 · Hosono
T-092 Smaller packings again at seven counts from n = 208 to 306, each certified two independent waysn = 208, 209, 228, 263, 272, 303, 306 · V3 C3 S3 · Couzo · superseded
T-094 Mixed rectangle-measure lower bounds verified at n = 67 and 84n = 67, 84 · V3 C3 S3 · Hosono
T-088 s(69) ≤ 8.82719465572975, Ellsworth's degree-38 packing, certified exactly from its picturen = 69 · V3 C3 S2 · Ellsworth · superseded
T-089 s(83) ≤ 9.63475764863195 and s(87) ≤ 9.83881526994915, Chang's packings, certified exactlyn = 83, 87 · V3 C3 S2 · Chang, Ellsworth · superseded
T-093 s(17) > 18641771/4000000 = 4.66044275n = 17 · V3 C3 S2 · Guzhou0806

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4 October 2026 · 5 results

T-083 s(n) ≥ 1/2 + √(n − ⌊√n⌋ + 1/4) for every nonsquare 8 ≤ n ≤ 324301 counts from 8 to 323 · V3 C3 S3 · Karakuş
T-084 s(k² − 1) = k for every integer k ≥ 3; k = 3…18 are the cases held here16 counts from 8 to 323 · V3 C3 S3 · Karakuş
T-085 Nagamochi 2005, Lemma 1 is false for every container with a > 3 and b > 2n = 10–324 · V3 C3 S3 · Karakuş; Vlasenko
T-086 s(k² − 2) = k for every integer k ≥ 2; k = 3…18 are the cases held here16 counts from 7 to 322 · V3 C3 S3 · Vlasenko
T-087 s(37) ≥ 5√3/2 + 2√2 − 1 and s(61) ≥ 7√3/2 + 2√2 − 1, by optimal piercingn = 37, 61 · V3 C1 S3 · Bašić, Slivková · superseded

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3 October 2026 · 17 results

T-081 s(k² − 4) = k for every integer k from 5 up; k = 5…18 are the cases held here14 counts from 21 to 320 · V0 C1 S4 · Daniel
T-066 s(59) = 8, by a mixed cover of points and grid-line segmentsn = 59 · V3 C3 S3 · Hosono
T-067 s(77) = 9, by a mixed cover extended from Daniel's s(60) covern = 77, 78 · V3 C3 S3 · Hosono
T-068 Rectangle-density lower bounds verified at 34 counts in n = 19…9534 counts from 19 to 95 · V3 C3 S3 · Hosono
T-069 Mixed rectangle-measure lower bounds replayed at n = 37, 65, 66, 90, 92n = 37, 65, 66, 90, 92 · V3 C3 S3 · Hosono
T-070 Rectangle-density lower bounds replayed at 25 counts in n = 29…9525 counts from 29 to 95 · V3 C3 S3 · Hosono · superseded
T-071 Mixed rectangle-measure lower bounds replayed at n = 84…87n = 84–87 · V3 C3 S3 · Hosono · superseded
T-072 Mixed rectangle-measure lower bound replayed at n = 76n = 76 · V3 C3 S3 · Hosono · superseded
T-073 Linear-measure lower bounds replayed at n = 83 and n = 101…105n = 83, 101–105 · V3 C3 S3 · Hosono
T-074 Rectangle-density lower bounds replayed at 31 counts in n = 19…9531 counts from 19 to 95 · V3 C3 S3 · Hosono
T-075 Mixed rectangle-measure lower bounds replayed at nine counts in n = 83…96n = 83, 85–88, 91–93, 96 · V3 C3 S3 · Hosono
T-076 Linear-measure lower bound replayed at n = 82n = 82 · V3 C3 S3 · Hosono
T-077 Rectangle-density lower bounds replayed at n = 20, 42 and 70n = 20, 42, 70 · V3 C3 S3 · Hosono · superseded
T-079 s(12) ≥ 15680000/3949423 = 3.9702002…, Daniel's points re-weightedn = 12 · V3 C3 S3 · Levy · superseded
T-080 Linear-measure lower bound replayed at n = 101…105n = 101–105 · V3 C3 S3 · Hosono
T-082 Mixed rectangle-measure lower bounds verified at 22 counts in n = 51…96n = 51, 52, 55, 58, 69–71, 73–76, 86–96 · V3 C3 S3 · Hosono
T-078 s(12) ≥ 31360/7901 = 3.9691178…, Daniel's certificate rescaled by 7902/7901n = 12 · V3 C3 S2 · Ryu · superseded

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1 October 2026 · 5 results

T-064 s(k² − 3) = k for every integer k from 6 up; k = 6…18 are the cases held here13 counts from 33 to 321 · V3 C3 S4 · Daniel
T-062 s(60) = 8, by a mixed cover of points and grid-line segmentsn = 60 · V3 C3 S3 · Daniel
T-063 s(61) = 8, by monotonicity from T-062n = 61 · V3 C3 S3 · Daniel
T-061 s(11) > 3875000000/999999999 = 3.875000003875…, 3.9e-9 above 31/8n = 11 · V3 C3 S2 · Wang, Li · superseded
T-065 s(17) ≤ 4.6755300936045509…, Bidwell's packing certified exactlyn = 17 · V3 C3 S2 · Overbeck

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30 September 2026 · 3 results · 1 change

T-060 Trump's eleven-square packing is globally optimaln = 11 · V3 C3 S5 · Ahmed
T-058 Rectangle-certificate ceiling α·UB(n) proved for n = 1..100; B·UB(n) on 64 grid rowsn = 1–100 · V3 C3 S2 · Hosono
T-059 Reported equality of 12028 n11 row minima reproduced by a complete bound replayn = 11 · V3 C3 S2 · Hosono
Ahmed Proves Trump’s Eleven-Square Packing OptimalS5 · PR #246

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29 September 2026 · 21 results

T-037 s(11) > 31/8 = 3.875n = 11 · V3 C3 S5 · Overbeck · superseded
T-047 s(11) ≥ 381/100; s(n) ≥ 1377/250 for n = 26…28; s(n) ≥ 571/100 for n = 29…31n = 11, 26–31 · V3 C3 S4 · Tokoharu · superseded
T-051 s(32) = 6n = 32 · V3 C3 S4 · Daniel
T-052 s(21) = 5, by a mixed cover of points and grid-line segmentsn = 21 · V3 C3 S4 · Daniel
T-038 s(17) > 461300/99853 = 4.6197910929…n = 17 · V3 C3 S3 · Overbeck · superseded
T-039 s(17) > 231001/50000 = 4.62002n = 17 · V3 C3 S3 · Guzhou0806 · superseded
T-040 s(17) > 232001/50000 = 4.64002n = 17 · V3 C3 S3 · Overbeck · superseded
T-041 s(17) > 466001/100000 = 4.66001n = 17 · V3 C3 S3 · Overbeck · superseded
T-043 s(17) > 116511/25000 = 4.66044n = 17 · V3 C3 S3 · Guzhou0806 · superseded
T-044 Weighted point lower bounds for ten counts in n = 26…72, plus seven from the same filesn = 26, 29, 39–41, 52–57, 68–73 · V3 C3 S3 · Hosono · superseded
T-045 Rectangle-density lower bounds replayed at 15 counts in n = 18…78n = 18–20, 26–28, 30–32, 40, 61, 75–78 · V3 C3 S3 · Hosono · superseded
T-046 Rectangle-density lower bounds verified at 48 counts in n = 18…9548 counts from 18 to 95 · V3 C3 S3 · Hosono · superseded
T-048 s(50) ≥ 37/5 = 7.4n = 50, 51 · V3 C3 S3 · Hosono
T-049 s(12) ≥ 15680/3951 = 3.9686155…n = 12 · V3 C3 S3 · Daniel · superseded
T-050 s(21) ≥ 5000/1001 = 4.995004995…n = 21 · V3 C3 S3 · Daniel · superseded
T-053 s(45) = 7, by a mixed cover of points and grid-line segmentsn = 45 · V3 C3 S3 · Daniel
T-056 Smaller packings for 49 counts from n = 68 to 307, each certified two independent ways49 counts from 68 to 307 · V3 C3 S3 · Couzo · superseded
T-057 s(211) ≤ 14.99796070496771500150 < 15, the first packing of 211 squares below the grid on recordn = 211 · V3 C3 S3 · de Winter · superseded
T-042 s(17) > 233009/50000 = 4.66018n = 17 · V3 C3 S2 · Guzhou0806 · superseded
T-054 s(45) = 7 by a second, point-only routen = 45 · V3 C3 S2 · Hosono
T-055 s(21) = 5 by a point-only routen = 21 · V3 C3 S2 · Hosono

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25 September 2026 · 3 results

T-034 s(21) ≥ 122/25 = 4.88n = 21 · V3 C3 S3 · Levy · superseded
T-035 Six-plus-five packings near Trump's tilt with side ≤ U_hi lie within rho of his posen = 11 · V3 C3 S3 · Levy
T-036 Trump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetryn = 11 · V3 C2 S3 · Levy

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23 September 2026 · 1 result

T-033 s(11) ≥ 955000√(2073600042893309449)/359341754646249 = 3.8269975…n = 11 · V3 C3 S3 · Levy · superseded

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21 September 2026 · 2 results

T-032 s(17) ≥ 461300/99999 = 4.61304613…, and beneath it Mira's s(17) ≥ 4613/1000n = 17 · V3 C3 S3 · Guzhou0806, Mira-acc · superseded
T-031 The octagon corner class (threshold 1/2) holds no eleven-square packing at side 96/25n = 11 · V3 C3 S2 · Levy · superseded

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20 September 2026 · 4 results

T-027 s(18) ≥ 467/100 = 4.67n = 18 · V3 C3 S3 · Levy · superseded
T-028 s(18) ≥ 187/40 = 4.675n = 18 · V3 C3 S3 · Levy · superseded
T-029 s(18) ≥ 1871/400 = 4.6775n = 18 · V3 C3 S3 · Levy · superseded
T-030 s(18) ≥ 4679/1000 = 4.679n = 18 · V3 C3 S3 · Levy · superseded

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10 September 2026 · 4 results

T-024 s(11) ≥ 3175000√(518400042893309449)/598960960743657 = 3.8166095…n = 11 · V3 C3 S5 · Levy · superseded
T-025 s(11) ≥ 191/50 = 3.82, by a threshold certificaten = 11 · V3 C3 S5 · Levy · superseded
T-026 s(11) ≥ 955000√(518400042893309449)/179696714646249 = 3.8264474…n = 11 · V3 C3 S5 · Levy · superseded
T-023 Conditional exclusion: no eleven-square packing in the four-owner branch at q = 96/25n = 11 · V3 C3 S3 · Levy

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6 September 2026 · 1 result

T-022 s(11) ≥ 38100√(8100042893309449)/899996306539 = 3.8100257…n = 11 · V3 C3 S5 · Levy · superseded

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5 September 2026 · 5 results

T-018 s(11) ≥ 381/100 = 3.81n = 11 · V3 C3 S5 · Levy · superseded
T-017 s(12) ≥ 99/25 = 3.96n = 12 · V3 C3 S4 · Levy · superseded
T-019 s(n) ≥ 459/100 = 4.59 for n = 17, 18, 19n = 17–19 · V3 C3 S4 · Levy · superseded
T-020 s(n) ≥ 24/5 = 4.80 for n = 19, 20, 21n = 19–21 · V3 C3 S4 · Levy · superseded
T-021 s(n) ≥ 97/20 = 4.85 for n = 20, 21n = 20, 21 · V3 C3 S3 · Levy · superseded

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3 September 2026 · 3 results

T-014 Goebel's n = 5 optimum is rigid at fixed side: its pose is an isolated feasible pointn = 5 · V3 C3 S3 · Levy
T-015 s(17) ≥ 22529/5000 = 4.5058n = 17 · V3 C3 S3 · Massaccesi · superseded
T-016 s(n) ≥ 22529/5000 for n = 18, 19, by monotonicity from T-015n = 18, 19 · V3 C3 S3 · Massaccesi · superseded

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1 September 2026 · 13 results

T-010 s(11) ≥ 2 + 4/√5, by a repair of Stromquist 2003's Figure 14 point setn = 11 · V3 C3 S4 · Levy · superseded
T-001 s(17) ≥ 4426213/1000000 = 4.426213, from a sixteen-point unavoidable setn = 17 · V3 C3 S3 · Levy · superseded
T-002 s(18) ≥ 4426213/1000000, by monotonicity from T-001n = 18 · V3 C3 S3 · Levy · superseded
T-004 Bentz 2010, Theorem 8 (s(46) ≥ 7) is correct as printed, machine-audited in fulln = 46 · V3 C3 S3 · Bentz
T-006 s(13) = 4n = 13 · V3 C3 S3 · Bentz; Daniel
T-007 s(n) ≥ min(⌈√n⌉, √(n - 2⌊√n⌋ + 1) + 1) for 4 ≤ n ≤ 324n = 4–324 · V0 C1 S3 · Nagamochi
T-008 s(46) = 7n = 46 · V3 C3 S3 · Bentz
T-009 s(29) ≤ 5.93383346267692918974…, by a Krawczyk interval certificaten = 29 · V3 C3 S3 · Levy · superseded
T-012 Goebel's n = 5 packing is second-order rigid at fixed siden = 5 · V3 C3 S3 · Levy
T-013 Goebel's n = 40 packing: seven verified first-order flexes, each refused at second ordern = 40 · V3 C3 S3 · Levy
T-003 The sixteen-point set's unavoidability ceiling lies in [4426213/1000000, 4427/1000)n = 17, 18 · V3 C3 S2 · Levy
T-005 Bentz 2010, Lemma 10 is false as printed and true as corrected to (1.74, 1)n = 13 · V3 C3 S2 · Levy
T-011 Trump's 1979 packing is exactly valid, so s(11) ≤ 3.877083590022814…n = 11 · V3 C3 S2 · Trump

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Changelog on GitHub