Square Packing: Bounds, Results and Best Packings

2026-10-09 published$s(132)\le 11.987099332245063\dots$ from Evan Daniel's record hunt132Daniel after Couzo, Chaoweeraprasit, LevyV0 C0upper boundrecordedT-1312026-10-08 publishedFive follow-up rational refinements from Francisco Couzo84, 86, 105, 175, 270Couzo after Xu, Daniel, Ellsworth, LevyV0 C0upper boundrecordedT-1302026-10-08 publishedEight complete rational refinements from Francisco Couzo105, 108, 127, 131, 155, 180, 228, 306Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, LevyV0 C0upper boundrecordedT-1282026-10-08 publishedFourteen rational refinements, seventeen complete source cases88, 108, 123, 129, 130, 153, 154, 179, 180, 199, 207–209, 236–239Gupta after Chaoweeraprasit, DanielV3 C3upper boundconfirmedT-1272026-10-08 publishedUndilated n51 construction over Q(sqrt2)51ry-xuV3 C3upper boundconfirmedT-1262026-10-08 publishedComplete rational construction reports at 25 counts51, 70, 84, 86, 88, 102, 103, 105, 108, 123, 126, 127, 129–131, 146, 153, 175, 179, 236, 258, 261, 263, 267, 295ry-xuV3 C3upper boundconfirmedT-1252026-10-07 publishedThree exact feasible upper bounds at n=266,270,272 from new source arrangements266, 270, 272Daniel after Levy, Ellsworth, Couzo, SteadV3 C3upper boundconfirmedT-1192026-10-07 published$s(n)\le S_n$ at nine counts, from the second SQUISH update’s new rational packings88, 108, 179, 180, 199, 207, 236, 263, 302Chaoweeraprasit after Ellsworth, Couzo and SQUISHV3 C3upper boundconfirmedT-1162026-10-07 published$s(n)\le S_n$ at twelve counts, from the SQUISH update’s new and smaller rational packings123, 126, 129, 154, 155, 179, 208, 237–239, 258, 263Chaoweeraprasit after Couzo and EllsworthV3 C3upper boundconfirmedT-1152026-10-07 published$s(n)\le S_n$ at ten counts, $n=108$ to 303, from new SQUISH rational packings108, 126, 129, 130, 154, 155, 180, 209, 238, 303Chaoweeraprasit after EllsworthV3 C3upper boundconfirmedT-1132026-10-06 establishedTrump's packing is the only optimal packing of eleven squares, up to symmetry11Levy after AhmedV3 C2uniquenessconfirmedT-1122026-10-06 published★Mixed rectangle-measure lower bound verified at $n=41$, on a declared net41wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1112026-10-06 published★Mixed rectangle-measure lower bound verified at $n=39$, on a declared net39wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1102026-10-06 published★Mixed rectangle-measure lower bound verified at $n=30$, on a declared net30wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1092026-10-06 published★Mixed rectangle-measure lower bound verified at $n=29$, on a declared net29wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1082026-10-06 published★Mixed rectangle-measure lower bound verified at $n=28$, on a declared net28wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1072026-10-06 published★Mixed rectangle-measure lower bound verified at $n=27$, on a declared net27wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1062026-10-06 published★Mixed rectangle-measure lower bound verified at $n=26$, on a declared net26wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1052026-10-06 published★Mixed rectangle-measure lower bound verified at $n=20$, on a declared net20wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1042026-10-06 published★Mixed rectangle-measure lower bound verified at $n=19$, on the finest declared net yet19wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1032026-10-06 published★Mixed rectangle-measure lower bound verified at $n=18$, on the finest declared net yet18wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-1022026-10-05 published★Mixed rectangle-measure lower bound verified at $n=66$66wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0972026-10-05 published★$s(12)\ge 7943/2000=3.9715$, Daniel's points dilated and re-weighted12squarepacker after Daniel, LevyV3 C3lower boundconfirmedT-0952026-10-05 published★Mixed rectangle-measure lower bounds verified at $n=67$ and $84$67, 84wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0942026-10-04 published★Mixed rectangle-measure lower bounds verified at 17 counts in $n=53\dots 95$53, 54, 58, 70–73, 76, 87–95wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0912026-10-04 published★Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$42–44, 51, 56, 57, 67, 69, 72, 75, 84, 86, 88, 93–96wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0902026-10-03 published★Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$51, 52, 55, 58, 69–71, 73–76, 86–96wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0822026-10-03 published$s(k^2-4)=k$ for every integer $k$ from 5 up; $k=5\dots 18$ are the cases held here21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320Daniel after Burns, MassaccesiV0 C1optimalityreviewedT-0812026-10-02 published★Linear-measure lower bound replayed at $n=101\dots 105$101–105wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0802026-10-02 published★Linear-measure lower bound replayed at $n=82$82wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0762026-10-02 published★Mixed rectangle-measure lower bounds replayed at nine counts in $n=83\dots 96$83, 85–88, 91–93, 96wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0752026-10-02 publishedLinear-measure lower bounds replayed at $n=83$ and $n=101\dots 105$83, 101–105wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0732026-10-01 published★Rectangle-density lower bounds replayed at 31 counts in $n=19\dots 95$19, 20, 26–31, 38–44, 53–58, 68–70, 74, 75, 88, 89, 93–95wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0742026-10-01 published★Mixed rectangle-measure lower bounds replayed at $n=37,65,66,90,92$37, 65, 66, 90, 92wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0692026-10-01 publishedRectangle-density lower bounds verified at 34 counts in $n=19\dots 95$19, 20, 26–31, 38–44, 53–56, 66, 68–70, 74–76, 86–90, 93–95wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0682026-10-01 published★$s(77)=9$, by a mixed cover extended from Daniel's $s(60)$ cover77, 78wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3optimalityconfirmedT-0672026-10-01 published★$s(59)=8$, by a mixed cover of points and grid-line segments59wand125 after Daniel, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3optimalityconfirmedT-0662026-09-30 published★$s(k^2-3)=k$ for every integer $k$ from 6 up; $k=6\dots 18$ are the cases held here33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321Daniel after Burns, MassaccesiV3 C3optimalityconfirmedT-0642026-09-29 published★$s(k^2-1)=k$ for every integer $k\ge 3$; $k=3\dots 18$ are the cases held here8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323KarakuşV3 C3optimalityconfirmedT-0842026-09-29 published★$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$8, 10–15, 17–24, 26–35, 37–48, 50–63, 65–80, 82–99, 101–120, 122–143, 145–168, 170–195, 197–224, 226–255, 257–288, 290–323KarakuşV3 C3lower boundconfirmedT-0832026-09-29 published★Trump's eleven-square packing is globally optimal11Ahmed after Levy, KleddamagV3 C3optimalityconfirmedT-0602026-09-28 published★$s(60)=8$, by a mixed cover of points and grid-line segments60Daniel after Burns, MassaccesiV3 C3optimalityconfirmedT-0622026-09-28 published★$s(50)\ge 37/5=7.4$50, 51wand125 after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, MassaccesiV3 C3lower boundconfirmedT-0482026-09-27 published★$s(45)=7$, by a mixed cover of points and grid-line segments45Daniel after Burns, MassaccesiV3 C3optimalityconfirmedT-0532026-09-27 published★$s(21)=5$, by a mixed cover of points and grid-line segments21Daniel after Burns, MassaccesiV3 C3optimalityconfirmedT-0522026-09-26 published★$s(32)=6$32Daniel after Burns, MassaccesiV3 C3optimalityconfirmedT-0512026-09-24 establishedTrump's pose is optimal among six-plus-five packings near its tilt, unique up to symmetry11LevyV3 C2restricted optimalityconfirmed superseded in part by T-060 and T-112T-0362026-09-24 establishedSix-plus-five packings near Trump's tilt with side $\le U_{hi}$ lie within rho of his pose11LevyV3 C3case exclusionconfirmedT-0352026-09-21 published$s(17)\le 4.6755300936045509\dots$, Bidwell's packing certified exactly17Kleddamag after Levy, Mira, Guzhou0806V3 C3upper boundconfirmedT-0652026-09-08 establishedConditional exclusion: no eleven-square packing in the four-owner branch at $q=96/25$11LevyV3 C3case exclusionconfirmed superseded in part by T-060T-0232026-09-04 published★$s(k^2-2)=k$ for every integer $k\ge 2$; $k=3\dots 18$ are the cases held here7, 14, 23, 34, 47, 62, 79, 98, 119, 142, 167, 194, 223, 254, 287, 322chelokotV3 C3optimalityconfirmedT-0862026-09-04 publishedNagamochi 2005, Lemma 1 is false for every container with $a>3$ and $b>2$10–324Karakuş; chelokotV3 C3correctionconfirmedT-0852026-09-03 establishedGoebel's $n=5$ optimum is rigid at fixed side: its pose is an isolated feasible point5LevyV3 C3rigidityconfirmedT-0142026-08-30 establishedGoebel's $n=40$ packing: seven verified first-order flexes, each refused at second order40LevyV3 C3rigidityconfirmedT-0132026-08-30 establishedGoebel's $n=5$ packing is second-order rigid at fixed side5LevyV3 C3rigidityconfirmedT-0122026-08-29 established$s(29)\le 5.933833\dots$, by a Krawczyk interval certificate29LevyV3 C3upper boundconfirmedT-009