T-056: Smaller packings for 49 counts from $n=68$$n=68$ to $307$$307$, each certified two independent ways

2005 published T-007 · 49 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Lower bound incomplete on these cases, superseded by T-073, T-080, T-083, T-180 (reported) and T-205

Nagamochi · Nagamochi 2005 · source · register

2026-09-04 published T-085 · 49 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

2026-09-22 published T-044 · case 68

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Lower bound confirmed superseded by T-205

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

2026-09-27 published T-046 · case 68

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

Lower bound recorded superseded by T-205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 this result · 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Upper bound confirmed superseded by T-098, T-127, T-136, T-146, T-173, T-174, T-182, T-186, T-190, T-197, T-218, T-219, T-222 and T-224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-070 · case 68

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

Lower bound confirmed superseded by T-205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 · case 68

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 · 49 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

2026-10-01 published T-068 · case 68

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-205

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

2026-10-01 published T-074 · case 68

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-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-073 · cases 102, 103, 105

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-080 · cases 102, 103, 105

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-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

2026-10-05 published T-098 · 46 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

2026-10-07 published T-113 · 7 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-174, T-186, T-197 and T-222

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

2026-10-07 published T-115 · 8 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-146, T-174, 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 · 6 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 270, 272

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

Upper bound confirmed on these cases, superseded by T-174 and T-224

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-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 · 8 of these cases

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-146, T-174 and T-218

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

2026-10-08 published T-127 · 12 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 · 6 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 · cases 105, 270

Five follow-up rational refinements from Couzo

Upper bound confirmed superseded by T-174 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 · 5 of these cases

Six exact rational certificates from Couzo

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

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

2026-10-09 published T-136 · 15 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 · 9 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-174, T-186 and T-224

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

2026-10-10 published T-140 · 5 of these cases

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-174, 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 · case 132

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

Upper bound confirmed superseded by T-174

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

2026-10-10 published T-144 · case 123

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 · 9 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-149 · cases 155, 156

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

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

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

2026-10-11 published T-151 · 8 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

2026-10-12 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, 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 · case 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

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

2026-10-12 published T-163 · cases 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 and T-182

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-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-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 · case 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 published T-174 · 45 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 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 and T-186

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

2026-10-13 published T-180 · 49 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 241, 302

Exact rational certificates at 211, 241 and 302 from Couzo

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

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, 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-186 · 6 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-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

2026-10-16 published T-193 · 7 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-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-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-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-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-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

2026-10-20 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

2026-10-22 published T-229 · case 105

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