T-180: Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

2005 published T-007 · 264 of these cases

s(n)≥min(⌈n​⌉,n−2⌊n​⌋+1​+1)$s(n)\ge min(\lceil \sqrt{n}\rceil ,\sqrt{n-2\lfloor \sqrt{n}\rfloor +1}+1)$ for $4\le n\le 324$$4\le n\le 324$

Lower bound incomplete

Nagamochi · Nagamochi 2005 · source · register

2018 published T-087 · case 61

s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

Lower bound reviewed superseded by T-063

Bašić, Slivková · Basic-Slivkova 2018 · source · register

2026-09-04 published T-085 · 264 of these cases

Nagamochi 2005, Lemma 1 is false for every container with $a>3$$a>3$ and $b>2$$b>2$

Correction confirmed

Karakuş; Vlasenko · Karakuş 2026 · chelokot Nagamochi counterexample 2026 · packet · register

2026-09-04 published T-086 · 11 of these cases

s(k2−2)=k$s(k^2-2)=k$ for every integer $k\ge 2$$k\ge 2$; $k=3\dots 18$$k=3\dots 18$ are the cases held here

Optimality confirmed

Vlasenko · chelokot Nagamochi counterexample 2026 · packet · source · register

2026-09-16 published T-057 · case 211

$s(211)\le 14.99796070496771500150<15$$s(211)\le 14.99796070496771500150<15$, the first packing of 211 squares below the grid on record

Upper bound confirmed superseded by T-188

de Winter · de Winter n211 2026-09-16 · packet · register

2026-09-22 published T-044 · 12 of these cases

Weighted point lower bounds for ten counts in $n=26\dots 72$$n=26\dots 72$, plus seven from the same files

Lower bound confirmed superseded by T-082, T-090, T-091 and T-205

Hosono after Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point bounds 2026 · packet · source · review · register

2026-09-24 published T-088 · case 69

$s(69)\le 8.82719465572975$$s(69)\le 8.82719465572975$, Ellsworth's degree-38 packing, certified exactly from its picture

Upper bound confirmed superseded by T-101 and T-155

Ellsworth after hmbelvedere, Cantrell, Schadt, Morandi · Ellsworth n69 2026-09-24 · packet · packet · register

2026-09-24 published T-089 · cases 83, 87

$s(83)\le 9.63475764863195$$s(83)\le 9.63475764863195$ and $s(87)\le 9.83881526994915$$s(87)\le 9.83881526994915$, Chang's packings, certified exactly

Upper bound confirmed superseded by T-101 and T-155

Chang, Ellsworth after Cantrell, Hajba, Stenlund, Bidwell; Chang after Ellsworth, Cantrell, Schadt, Hajba, DeVincentis · Chang n83 2026-09-24 · Chang n87 2026-09-24 · packet · packet · register

2026-09-27 published T-045 · 5 of these cases

Rectangle-density lower bounds replayed at 15 counts in $n=18\dots 78$$n=18\dots 78$

Lower bound confirmed superseded by T-063, T-064, T-067, T-090 and T-091

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register

2026-09-27 published T-046 · 31 of these cases

Rectangle-density lower bounds reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

Lower bound recorded superseded by T-062, T-063, T-064, T-066, T-067, T-082, T-090, T-091, T-094, T-097 and T-205

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register

2026-09-27 published T-056 · 49 of these cases

Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

Upper bound confirmed superseded by T-098, T-127, T-136, T-146, T-173, T-174, T-182, T-186, T-190, T-197, T-218, T-219, T-222 and T-224

Couzo · franciscouzo square-packing 2026-09-27 · packet · register

2026-09-28 published T-048 · cases 50, 51

$s(50)\ge 37/5=7.4$$s(50)\ge 37/5=7.4$

Lower bound confirmed

Hosono after Daniel, Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 point and mixed bounds 2026-09-28 · packet · source · review · register

2026-09-28 published T-062 · case 60

$s(60)=8$$s(60)=8$, by a mixed cover of points and grid-line segments

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · source · review · register

2026-09-28 published T-063 · case 61

$s(61)=8$$s(61)=8$, by monotonicity from T-062

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-09-28 published T-070 · 18 of these cases

Rectangle-density lower bounds replayed at 25 counts in $n=29\dots 95$$n=29\dots 95$

Lower bound confirmed superseded by T-062, T-066, T-082, T-090, T-091, T-094 and T-205

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-09-28 · packet · packet · source · review · register

2026-09-29 published T-058 · 48 of these cases

Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

Method limit confirmed

Hosono after Tokoharu, Daniel · wand125 tools 2026 · packet · register

2026-09-29 published T-083 · 264 of these cases

s(n)≥1/2+n−⌊n​⌋+1/4​$s(n)\ge 1/2+\sqrt{n-\lfloor \sqrt{n}\rfloor +1/4}$ for every nonsquare $8\le n\le 324$$8\le n\le 324$

Lower bound confirmed

Karakuş · Karakuş 2026 · source · register

2026-09-29 published T-084 · 11 of these cases

s(k2−1)=k$s(k^2-1)=k$ for every integer $k\ge 3$$k\ge 3$; $k=3\dots 18$$k=3\dots 18$ are the cases held here

Optimality confirmed

Karakuş · Karakuş 2026 · source · register

2026-09-30 published T-064 · 11 of these cases

s(k2−3)=k$s(k^2-3)=k$ for every integer $k$$k$ from 6 up; $k=6\dots 18$$k=6\dots 18$ are the cases held here

Optimality confirmed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-10-01 published T-066 · case 59

$s(59)=8$$s(59)=8$, by a mixed cover of points and grid-line segments

Optimality confirmed

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

2026-10-01 published T-067 · cases 77, 78

$s(77)=9$$s(77)=9$, by a mixed cover extended from Daniel's $s(60)$$s(60)$ cover

Optimality confirmed

Hosono after Daniel, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 exact covers 2026-10-01 · packet · source · review · register

2026-10-01 published T-068 · 19 of these cases

Rectangle-density lower bounds verified at 34 counts in $n=19\dots 95$$n=19\dots 95$

Lower bound confirmed on these cases, superseded by T-082, T-090, T-091, T-097 and T-205

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · source · review · register

2026-10-01 published T-069 · cases 65, 66, 90, 92

Mixed rectangle-measure lower bounds replayed at $n=37,65,66,90,92$$n=37,65,66,90,92$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register

2026-10-01 published T-074 · 16 of these cases

Rectangle-density lower bounds replayed at 31 counts in $n=19\dots 95$$n=19\dots 95$

Lower bound confirmed on these cases, superseded by T-082, T-090, T-091 and T-205

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-01 · packet · packet · source · review · register

2026-10-02 published T-071 · cases 84, 85, 86, 87

Mixed rectangle-measure lower bounds replayed at $n=84\dots 87$$n=84\dots 87$

Lower bound confirmed superseded by T-090, T-091, T-206 and T-207

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · packet · source · review · register

2026-10-02 published T-072 · case 76

Mixed rectangle-measure lower bound replayed at $n=76$$n=76$

Lower bound confirmed superseded by T-091

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds n76 2026-10-02 · packet · packet · source · review · register

2026-10-02 published T-073 · 6 of these cases

Linear-measure lower bounds replayed at $n=83$$n=83$ and $n=101\dots 105$$n=101\dots 105$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

2026-10-02 published T-075 · 9 of these cases

Mixed rectangle-measure lower bounds replayed at nine counts in $n=83\dots 96$$n=83\dots 96$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds afternoon 2026-10-02 · packet · source 1 · source 2 · review · register

2026-10-02 published T-076 · case 82

Linear-measure lower bound replayed at $n=82$$n=82$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n82 2026-10-02 · packet · source · review · register

2026-10-02 published T-077 · case 70

Rectangle-density lower bounds replayed at $n=20$$n=20$, 42 and 70

Lower bound confirmed superseded by T-091

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026-10-02 · packet · packet · source 1 · source 2 · review 1 · review 2 · register

2026-10-02 published T-080 · 5 of these cases

Linear-measure lower bound replayed at $n=101\dots 105$$n=101\dots 105$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear certificates 2026-10-02 · packet · source · review · register

2026-10-03 published T-081 · 11 of these cases

s(k2−4)=k$s(k^2-4)=k$ for every integer $k$$k$ from 5 up; $k=5\dots 18$$k=5\dots 18$ are the cases held here

Optimality reviewed

Daniel after Burns, Massaccesi · evand square-packing 2026-10-03 · packet · packet · source · review · register

2026-10-03 published T-082 · 22 of these cases

Mixed rectangle-measure lower bounds verified at 22 counts in $n=51\dots 96$$n=51\dots 96$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-03 · packet · packet · source · review · register

2026-10-03 published T-092 · 7 of these cases

Smaller packings again at seven counts from $n=208$$n=208$ to $306$$306$, each certified two independent ways

Upper bound confirmed superseded by T-174 and T-186

Couzo · franciscouzo square-packing 2026-10-03 · packet · register

2026-10-04 published T-090 · 14 of these cases

Mixed rectangle-measure lower bounds verified at 17 counts in $n=42\dots 96$$n=42\dots 96$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-04 · packet · source · review · register

2026-10-04 published T-091 · 17 of these cases

Mixed rectangle-measure lower bounds verified at 17 counts in $n=53\dots 95$$n=53\dots 95$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds evening 2026-10-04 · packet · source · review · register

2026-10-05 published T-094 · cases 67, 84

Mixed rectangle-measure lower bounds verified at $n=67$$n=67$ and $84$$84$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-05 · packet · source · review · register

2026-10-05 published T-097 · case 66

Mixed rectangle-measure lower bound verified at $n=66$$n=66$

Lower bound confirmed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds finer net 2026-10-05 · packet · source · review · register

2026-10-05 published T-098 · 48 of these cases

Exact optima of 48 known-best packings: s(n) ≤ S', 3.5e-13 to 5.0e-11 below each printed side

Upper bound confirmed

Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

2026-10-05 published T-101 · 73 of these cases

Exact certificates of 77 catalogue packings: s(n) ≤ S', 1.2e-16 to 9.8e-15 above each side

Upper bound confirmed

Daniel after Couzo, de Winter, Ellsworth, Levy · evand exact optima 2026-10-05 · packet · register

2026-10-07 published T-113 · 10 of these cases

s(n)≤Sn​$s(n)\le S_n$ at ten counts, $n=108$$n=108$ to 303, from new SQUISH rational packings

Upper bound confirmed superseded by T-139, T-174, T-186, T-197 and T-222

Chaoweeraprasit after Ellsworth · SQUISH ten packings 2026-10-07 · packet · register

2026-10-07 published T-114 · case 153

$s(153)\le 7250614903299225/562949953421312$, from a SQUISH rational packing

Upper bound confirmed superseded by T-173

Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register

2026-10-07 published T-115 · 12 of these cases

s(n)≤Sn​$s(n)\le S_n$ at twelve counts, from the SQUISH update’s new and smaller rational packings

Upper bound confirmed superseded by T-139, T-146, T-174, T-186, T-190, T-197 and T-222

Chaoweeraprasit after Couzo and Ellsworth · SQUISH update 2026-10-07 · packet · register

2026-10-07 published T-116 · 9 of these cases

s(n)≤Sn​$s(n)\le S_n$ at nine counts, from the second SQUISH update’s new rational packings

Upper bound confirmed superseded by T-127, T-173, T-174, T-186 and T-218

Chaoweeraprasit after Ellsworth, Couzo and SQUISH · SQUISH second update 2026-10-07 · packet · register

2026-10-07 published T-117 · cases 105, 292

Exact rational ceiling refinements at $n=105$$n=105$, 292

Upper bound confirmed superseded by T-146 and T-174

Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 2026-10-07 · packet · register

2026-10-07 published T-118 · case 68

Exact rational ceiling refinements at $n=68$$n=68$

Upper bound confirmed superseded by T-174

Rehwaldt after Couzo and earlier contributors · Rehwaldt n68 refinement 2026-10-07 · packet · register

2026-10-07 published T-119 · cases 266, 270, 272

Three exact feasible upper bounds at n=266,270,272 from new source arrangements

Upper bound confirmed

Daniel after Levy, Ellsworth, Couzo, Stead · Daniel new arrangements 2026-10-07 · packet · register

2026-10-07 published T-120 · case 102

Reported s(102) ≤ alpha_102 with a degree-8 exact side form

Upper bound recorded superseded by T-146

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-121 · case 106

Reported s(106) ≤ alpha_106 with a degree-32 exact side form

Upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-122 · case 152

Reported s(152) ≤ alpha_152 with a degree-40 exact side form

Upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-123 · case 177

Reported s(177) ≤ alpha_177 with a degree-32 exact side form

Upper bound recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-124 · 131 of these cases

Reported non-strict local minima for 178 source configurations

Restricted optimality recorded

Daniel after Couzo · Daniel exact and local reports 2026 · packet · register

2026-10-07 published T-129 · cases 105, 130, 292

Three dated Daniel certificate reports with historical source custody

Upper bound recorded superseded by T-146 and T-174

Daniel after Couzo, Levy · Daniel dated certificates 105 and 130 2026-10-07 · Daniel dated certificate 292 2026-10-07 · packet · packet · packet · register

2026-10-08 published T-125 · 25 of these cases

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-126, T-127, T-134, T-139, T-146, T-173, T-174, T-186, T-187 and T-218

Xu · ry-xu square packing 2026 · packet · register

2026-10-08 published T-126 · case 51

Undilated n51 construction over Q(sqrt2)

Upper bound confirmed

Xu · ry-xu square packing 2026 · packet · register

2026-10-08 published T-127 · 17 of these cases

Fourteen rational refinements, seventeen complete source cases

Upper bound confirmed

Gupta after Chaoweeraprasit, Daniel · Gupta rational refinements 2026-10-08 · packet · register

2026-10-08 published T-128 · 8 of these cases

Eight complete rational refinements from Couzo

Upper bound confirmed superseded by T-174 and T-197

Couzo after Xu, Chaoweeraprasit, Gupta, Ellsworth, Daniel, Levy · Couzo exact refinements 2026-10-08 · packet · register

2026-10-08 published T-130 · 5 of these cases

Five follow-up rational refinements from Couzo

Upper bound confirmed superseded by T-174, T-186 and T-224

Couzo after Xu, Daniel, Ellsworth, Levy · Couzo follow-up refinements 2026-10-08 · packet · register

2026-10-09 published T-131 · case 132

$s(132)\le 11.987099332245063\dots$$s(132)\le 11.987099332245063\dots$ from Daniel's record hunt

Upper bound confirmed superseded by T-174

Daniel after Couzo, Chaoweeraprasit, Levy · Daniel record hunt 2026-10-09 · packet · register

2026-10-09 published T-134 · 6 of these cases

Six exact rational certificates from Couzo

Upper bound confirmed

Couzo after Polkovskyi, Chaoweeraprasit, Gupta, Xu, Daniel, Ellsworth, Levy · Couzo exact certificates 2026-10-09 · packet · register

2026-10-09 published T-136 · 17 of these cases

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed

Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 2026-10-09 · packet · register

2026-10-09 published T-138 · 16 of these cases

Exact witnesses at Mishapolk's printed ceilings, the smallest known at 103 and 258 when registered

Upper bound confirmed superseded by T-134, T-146, T-174, T-186 and T-224

Polkovskyi after Stenlund, Friedman, Ellsworth, Chaoweeraprasit, Couzo, Xu, Levy · Mishapolk decimal poses 2026-10-09 · packet · register

2026-10-09 published T-139 · case 126

A more precise certificate of Xu's 126-square arrangement, from Daniel

Upper bound confirmed

Daniel after Xu, Chaoweeraprasit · Daniel trio126 2026-10-09 · packet · register

2026-10-10 published T-135 · case 308

A packing of 308 squares below the grid, from Fang

Upper bound confirmed superseded by T-221

Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead · Fang two-wedge certificates 2026-10-10 · packet · register

2026-10-10 published T-137 · case 70

An exact rational refinement of Xu's packing of 70 squares

Upper bound confirmed superseded by T-146

Deleeuw after Xu, Levy · Deleeuw n70 refinement 2026-10-10 · packet · register

2026-10-10 published T-140 · 6 of these cases

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-174, T-186, T-190 and T-224

Couzo after Chaoweeraprasit, Polkovskyi, Xu, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-10 · packet · register

2026-10-10 published T-141 · cases 132, 308

Exact rational certificates at 132 and 308 from Daniel's third record hunt

Upper bound confirmed superseded by T-174 and T-221

Daniel after Fang, Arslanov, Mustafin, Shangitbayev, Couzo, Levy · Daniel record hunt 3 2026-10-10 · packet · register

2026-10-10 published T-144 · 5 of these cases

Linear-measure lower bound $s(n)\ge 563/50=11.26$$s(n)\ge 563/50=11.26$ for $n=122\dots 126$$n=122\dots 126$, reported

Lower bound recorded

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n122 2026-10-10 · packet · register

2026-10-10 published T-146 · 15 of these cases

Exact rational certificates at fifteen counts from Daniel's regularized record lists

Upper bound confirmed

Daniel after Xu, Chaoweeraprasit, Polkovskyi, Levy · Daniel regularized lists 2026-10-10 · packet · register

2026-10-11 published T-147 · case 207

A packing of 207 squares from two placed patches of Xu's packing of 70

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n207 certificate 2026-10-11 · packet · register

2026-10-11 published T-148 · case 153

A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register

2026-10-11 published T-149 · cases 155, 156, 211

New packings of 155, 156 and 211 squares, each with an exact rational certificate

Upper bound confirmed superseded by T-186, T-188 and T-197

Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register

2026-10-11 published T-150 · cases 146, 261

Exact rational refinements of Xu's packings of 146 and 261 squares

Upper bound confirmed superseded by T-174 and T-187

Barnett after Xu, Daniel, Levy · Barnett refinements 2026-10-11 · packet · register

2026-10-11 published T-151 · 9 of these cases

Nine exact rational packings from SQUISH's fourth request

Upper bound confirmed superseded by T-174, T-186, T-219 and T-224

Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · packet · register

2026-10-11 published T-152 · cases 304, 305

Exact rational refinements of SQUISH's Mondrian packings of 304 and 305 squares

Upper bound confirmed superseded by T-174 and T-186

Couzo after Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-11 · packet · register

2026-10-11 published T-159 · case 102

Another arrangement of Xu's design for 102 squares, certified exactly from its decimal layout

Upper bound confirmed superseded by T-146

Calle after Xu, Levy · Calle n102 2026-10-12 · packet · register

2026-10-11 published T-185 · case 175

A tighter packing of 175 squares from Couzo, superseded by Ostashev's composition

Upper bound confirmed superseded by T-186

Couzo after Xu, Daniel, Ellsworth, Levy · Couzo n175 2026-10-11 · packet · register

2026-10-12 established T-155 · 54 of these cases

Exact regularized sides at 59 counts, 5.4e-20 to 9.9e-15 below the ceilings they replace

Upper bound confirmed

Levy after Daniel · register

2026-10-12 published T-158 · cases 175, 232, 264

New packings of 175, 232 and 264 squares built on a new 38-square rectangle, each certified exactly

Upper bound confirmed superseded by T-186

Ostashev after Nakajima, Ellsworth, Schadt, Friedman, de Winter, Daniel, Levy, Xu · Ostashev records 2026-10-12 · packet · register

2026-10-12 published T-160 · case 154

A smaller packing of 154 squares from Daniel's size-chaining explorer, certified exactly

Upper bound confirmed superseded by T-222

Daniel after Couzo, Levy · Daniel n154 2026-10-12 · packet · register

2026-10-12 published T-161 · cases 156, 176, 237, 271

Exact rational certificates at 156, 176, 237 and 271 from Couzo

Upper bound confirmed superseded by T-186, T-190 and T-219

Couzo after Ostashev, Chaoweeraprasit, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-12 · packet · register

2026-10-12 published T-162 · cases 129, 204, 303

New packings of 129, 204 and 303 squares, a cut-out and two compositions, each certified exactly

Upper bound confirmed superseded by T-186 and T-225

Ostashev after Nakajima, Chaoweeraprasit, Daniel, Xu, Levy · Ostashev compositions 2026-10-12 · packet · register

2026-10-12 published T-163 · cases 211, 241, 273, 307

s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang

Upper bound reviewed superseded by T-174, T-182 and T-188

Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · register

2026-10-12 published T-164 · case 302

A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks

Upper bound confirmed superseded by T-186

PingYou Ltd after Xu, Daniel, Levy · PingYou n302 certificate 2026-10-12 · packet · register

2026-10-12 published T-165 · case 176

A tighter certificate at 176 from Couzo, published with no issue

Upper bound confirmed superseded by T-186

Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register

2026-10-12 published T-166 · case 153

A tighter rational realization of PingYou Ltd's packing of 153 squares

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register

2026-10-12 published T-167 · case 207

A tighter rational realization of PingYou Ltd's packing of 207 squares

Upper bound confirmed superseded by T-173

PingYou Ltd after Xu, Daniel, Levy · PingYou n207 refinement 2026-10-12 · packet · register

2026-10-12 published T-168 · case 146

A further exact refinement of Xu's packing of 146 squares, after Barnett's

Upper bound confirmed superseded by T-187

PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n146 refinement 2026-10-12 · packet · register

2026-10-12 published T-169 · case 261

A further exact refinement of Xu's packing of 261 squares, after Barnett's

Upper bound confirmed superseded by T-174

PingYou Ltd after Xu, Barnett, Daniel, Levy · PingYou n261 refinement 2026-10-12 · packet · register

2026-10-12 published T-170 · case 105

An exact contact refinement of Xu's packing of 105 squares, after Couzo's

Upper bound confirmed superseded by T-174

PingYou Ltd after Xu, Couzo, Daniel, Levy · PingYou n105 refinement 2026-10-12 · packet · register

2026-10-12 published T-171 · case 272

An exact contact refinement of SQUISH's packing of 272 squares

Upper bound confirmed superseded by T-174

PingYou Ltd after Chaoweeraprasit, Daniel, Levy · PingYou n272 refinement 2026-10-12 · packet · register

2026-10-12 published T-172 · case 154

A packing of 154 squares from Couzo, continuing Daniel's arrangement

Upper bound confirmed superseded by T-222

Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · packet · register

2026-10-12 published T-173 · cases 153, 207

Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

Upper bound confirmed

Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register

2026-10-13 established T-156 · case 290

$s(290)\le 17.5355339059327376\dots$$s(290)\le 17.5355339059327376\dots$, by monotonicity from T-155, with its own exact certificate

Upper bound confirmed

Levy after Daniel · register

2026-10-13 published T-174 · 68 of these cases

Tighter certificates at 68 counts from PingYou Ltd, of others' packings and its own

Upper bound confirmed

PingYou Ltd after Friedman, Ellsworth, Daniel, Xu, Barnett, Couzo, Ostashev, Chaoweeraprasit, Gupta, Fang, Polkovskyi, Levy · PingYou PR 574 certificates 2026-10-13 · packet · register

2026-10-13 published T-175 · cases 211, 241, 304

New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly

Upper bound confirmed superseded by T-182, T-186 and T-188

Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register

2026-10-13 published T-179 · 5 of these cases

Linear-measure lower bound $s(n)\ge 306/25=12.24$$s(n)\ge 306/25=12.24$ for $n=145\dots 149$$n=145\dots 149$, reported

Lower bound reviewed

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 linear n145 2026-10-13 · packet · register

2026-10-13 published T-180 this result · 264 of these cases

Closed-form lower bound $s(N)\ge a(N)$$s(N)\ge a(N)$ for every non-square $N\ge 49$$N\ge 49$, reported

Lower bound reviewed

Hosono · wand125 closed form 2026-10-14 · packet · packet · register

2026-10-13 published T-181 · cases 211, 241, 302

Exact rational certificates at 211, 241 and 302 from Couzo

Upper bound confirmed superseded by T-182, T-186 and T-188

Couzo after Daniel, Ostashev, Fang, Ellsworth, Levy; Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo issue-583 certificates 2026-10-13 · Couzo certificates 2026-10-13 · packet · packet · register

2026-10-13 published T-182 · cases 154, 211, 241

Exact rational certificates at 154, 211 and 241 from Couzo

Upper bound confirmed

Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register

2026-10-13 published T-183 · case 237

Polkovskyi's certificate for 237 squares, the smallest known side there when registered

Upper bound confirmed superseded by T-190

Polkovskyi after Friedman, Daniel, Xu, Chaoweeraprasit, Barnett, Couzo, Ostashev, Ellsworth, Fang, Levy · Mishapolk records 2026-10-14 · packet · register

2026-10-14 published T-184 · case 76

$s(76)=9$$s(76)=9$, by a mixed cover of points and grid-line segments

Optimality reviewed

Greff after Daniel, Hosono · bengreff s76 certificate 2026-10-14 · packet · source · review · register

2026-10-14 published T-186 · 12 of these cases

Exact certificates at twelve counts from Polkovskyi, tightening the smallest reported there

Upper bound confirmed

Polkovskyi after Ostashev, Couzo, PingYou Ltd, Levy · Mishapolk KKT refinements 2026-10-14 · packet · register

2026-10-14 published T-187 · case 146

A smaller packing of 146 squares from Daniel's chain search, started from Xu's packing

Upper bound confirmed

Daniel after Xu · Daniel n146 2026-10-14 · packet · register

2026-10-14 published T-188 · case 211

A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangement

Upper bound confirmed

Couzo after Ostashev, Daniel, Levy · Couzo verified bounds 2026-10-14 · packet · register

2026-10-14 published T-189 · case 237

Couzo's refinement of Polkovskyi's 237, and fifteen beyond-horizon bounds

Upper bound confirmed superseded by T-190

Couzo after Polkovskyi, Fang, Daniel, Ellsworth, Levy · Couzo certificates 2026-10-14 · packet · register

2026-10-15 published T-190 · case 237

A tighter exact certificate for 237 squares from Couzo, below his issue-607 certificate

Upper bound confirmed

Couzo after Polkovskyi, Chaoweeraprasit, Daniel, Levy · Couzo n237 2026-10-15 · packet · register

2026-10-15 published T-191 · case 95

$s(95)=10$$s(95)=10$, by a mixed cover of points and grid-line segments

Optimality reviewed

Greff after Daniel, Hosono · bengreff s95 certificate 2026-10-15 · packet · source · review · register

2026-10-16 published T-193 · 16 of these cases

Exact algebraic sides at sixteen counts from Daniel, of packings the record credits

Upper bound confirmed

Daniel after Xu, Chaoweeraprasit, Ostashev, Couzo, PingYou Ltd, Gupta, Polkovskyi, Levy · Daniel exact current records 2026-10-16 · packet · register

2026-10-16 published T-204 · case 51

Mixed rectangle-measure lower bound $s(51)\ge 3011/400$$s(51)\ge 3011/400$ reported, on a 2,073-direction declared net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-205 · case 68

Mixed rectangle-measure lower bound $s(68)\ge 853/100$$s(68)\ge 853/100$ verified, on the 201-direction net

Lower bound confirmed

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

2026-10-16 published T-206 · case 84

Mixed rectangle-measure lower bound $s(84)\ge 3769/400$$s(84)\ge 3769/400$ verified, on the 201-direction net

Lower bound confirmed

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

2026-10-16 published T-207 · case 85

Mixed rectangle-measure lower bound $s(85)\ge 379/40$$s(85)\ge 379/40$ verified, on the 201-direction net

Lower bound confirmed

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

2026-10-16 published T-208 · case 86

Mixed rectangle-measure lower bound $s(86)\ge 3811/400$$s(86)\ge 3811/400$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-209 · case 88

Mixed rectangle-measure lower bound $s(88)\ge 3853/400$$s(88)\ge 3853/400$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-210 · case 89

Mixed rectangle-measure lower bound $s(89)\ge 1937/200$$s(89)\ge 1937/200$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-211 · case 91

Mixed rectangle-measure lower bound $s(91)\ge 783/80$$s(91)\ge 783/80$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-212 · case 90

Mixed rectangle-measure lower bound $s(90)\ge 489/50$$s(90)\ge 489/50$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-213 · case 92

Mixed rectangle-measure lower bound $s(92)\ge 493/50$$s(92)\ge 493/50$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-214 · case 106

Mixed rectangle-measure lower bound $s(106)\ge 10503/1000$$s(106)\ge 10503/1000$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-215 · case 107

Mixed rectangle-measure lower bound $s(107)\ge 529/50$$s(107)\ge 529/50$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-216 · case 108

Mixed rectangle-measure lower bound $s(108)\ge 531/50$$s(108)\ge 531/50$ verified, on the 201-direction net

Lower bound confirmed

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · source · review · register

2026-10-16 published T-217 · case 110

Mixed rectangle-measure lower bound $s(110)\ge 1073/100$$s(110)\ge 1073/100$ reported, on the 201-direction net

Lower bound recorded

Hosono after Tokoharu and Levy · wand125 fine-net lower bounds 2026-10-16 · packet · register

2026-10-16 published T-218 · case 236

Daniel's smaller n236 packing, after Chaoweeraprasit's composition, certified exactly

Upper bound confirmed

Daniel after Chaoweeraprasit, Levy · Daniel n236 2026-10-16 · packet · register

2026-10-17 published T-196 · cases 227, 229, 260

de Winter reports three smaller rational upper bounds at 227, 229 and 260

Upper bound confirmed superseded by T-225

de Winter after Levy · de Winter upper bounds 2026-10-17 · packet · register

2026-10-17 published T-197 · case 155

Daniel's smaller n155 packing, started from Ostashev's packing

Upper bound confirmed

Daniel after Ostashev, Levy · Daniel n155 explorer 2026-10-17 · packet · register

2026-10-17 published T-198 · case 229

Daniel's opening n229 dilation certificate, after Couzo and de Winter

Upper bound confirmed superseded by T-225

Daniel after Couzo, de Winter, Levy · Daniel n229 dilation 2026-10-17 · packet · register

2026-10-17 published T-219 · case 271

Daniel's smaller n271 packing, continuing Couzo's arrangement, certified exactly

Upper bound confirmed

Daniel after Couzo, Chaoweeraprasit, Levy · Daniel n271 2026-10-17 · packet · register

2026-10-17 published T-226 · case 204

Couzo's 204 of #670, below Polkovskyi's T-186, since superseded by his own #675 and #678

Upper bound recorded superseded by T-225

Couzo after Polkovskyi, Daniel, Levy · Couzo issue 670 2026-10-17 · packet · packet · register

2026-10-17 published T-227 · cases 227, 229

Couzo's refinements of de Winter's 227 and 229, below T-196 at both

Upper bound recorded superseded by T-225

Couzo after de Winter, Daniel, Levy · Couzo issue 672 2026-10-17 · packet · packet · register

2026-10-18 published T-199 · case 229

Daniel's smaller n229 update, replacing the opening certificate

Upper bound confirmed superseded by T-225

Daniel after Couzo, de Winter, Levy · Daniel n229 dilation update 2026-10-18 · packet · register

2026-10-18 published T-223 · cases 204, 260

Couzo's smaller packings at 204 and 260, the two in the corpus of issue 678's 85 bounds

Upper bound confirmed superseded by T-225

Couzo after de Winter, Polkovskyi, Daniel, Fang, PingYou Ltd, Levy · Couzo issue 678 2026-10-18 · packet · register

2026-10-18 published T-228 · case 204

Couzo's 204 of #675, below his #670, since superseded by his #678

Upper bound recorded superseded by T-225

Couzo after Polkovskyi, Daniel, Levy · Couzo issue 675 2026-10-18 · packet · packet · register

2026-10-19 published T-220 · case 183

Daniel's finite sub-grid upper bound for 183 squares, after Couzo's n182 packing

Upper bound confirmed

Daniel after Couzo; register seed data from Levy · Daniel n183 subgrid 2026-10-19 · packet · register

2026-10-19 published T-221 · case 308

Daniel's smaller n308 packing, started from Chaoweeraprasit's 307 packing, certified exactly

Upper bound confirmed

Daniel after Chaoweeraprasit, Levy · Daniel n308 2026-10-19 · packet · register

2026-10-20 published T-222 · case 154

Daniel's smaller n154 packing, continuing Couzo's #586 arrangement, certified exactly

Upper bound confirmed

Daniel after Couzo, Ostashev, Levy · Daniel n154 sweep 2026-10-20 · packet · packet · register

2026-10-20 published T-224 · case 270

Daniel's smaller n270 packing, another arrangement found from Chaoweeraprasit's, certified exactly

Upper bound confirmed

Daniel after Chaoweeraprasit, Levy · Daniel n270 sweep 2026-10-20 · packet · register

2026-10-20 published T-225 · cases 204, 227, 229, 260

Daniel's solved, regularized views at 204, 227, 229 and 260, each the smallest side reported there

Upper bound confirmed

Daniel after Couzo, de Winter, Polkovskyi, Levy · Daniel regularized lists 2026-10-20 · packet · register

2026-10-22 published T-229 · cases 105, 204

Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225

Upper bound recorded

Polkovskyi after Couzo, Muglich, Daniel, Xu, PingYou Ltd, Levy · Mishapolk issue 719 2026-10-22 · packet · register