T-115: s(n)≤Sn​$s(n)\le S_n$ at twelve counts, from the SQUISH update’s new and smaller rational packings

2005 published T-007 · 12 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Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)

Nagamochi · Nagamochi 2005 · source · register

2026-09-04 published T-085 · 12 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

2026-09-27 published T-056 · 8 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Upper bound confirmed superseded by T-146, T-174, T-190, T-197 and T-222

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

2026-09-29 published T-083 · 12 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

2026-10-03 published T-092 · cases 208, 263

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Upper bound confirmed superseded by T-174

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

2026-10-05 published T-098 · 9 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Upper bound confirmed on these cases, superseded by T-139, T-146, T-174, T-190, T-197 and T-222

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

2026-10-05 published T-101 · cases 129, 179, 258

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

Upper bound confirmed on these cases, superseded by T-146, T-174 and T-186

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

2026-10-07 published T-113 · 5 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Upper bound confirmed superseded by T-139, T-174, T-186, T-197 and T-222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-115 this result · 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 · cases 179, 263

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-174

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

2026-10-08 published T-125 · 6 of these cases

Complete rational construction reports at 25 counts

Upper bound confirmed superseded by T-139, T-146, T-174 and T-186

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

2026-10-08 published T-127 · 8 of these cases

Fourteen rational refinements, seventeen complete source cases

Upper bound confirmed on these cases, superseded by T-146, T-174, T-186, T-190 and T-222

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

2026-10-08 published T-128 · case 155

Eight complete rational refinements from Couzo

Upper bound confirmed superseded by T-197

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

2026-10-09 published T-134 · cases 237, 263

Six exact rational certificates from Couzo

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

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

2026-10-09 published T-136 · cases 154, 237, 263

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed on these cases, superseded by T-174, T-190 and T-222

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

2026-10-09 published T-138 · case 258

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

Upper bound confirmed superseded by T-146

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-140 · case 237

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-190

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

2026-10-10 published T-144 · cases 123, 126

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 · cases 123, 129, 258, 263

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-149 · case 155

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

Upper bound confirmed superseded by T-197

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

2026-10-11 published T-151 · cases 238, 239

Nine exact rational packings from SQUISH's fourth request

Upper bound confirmed superseded by T-174

Chaoweeraprasit after Couzo, Gupta, Ellsworth, Friedman, Levy · SQUISH fourth request 2026-10-11 · 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 · case 237

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

Upper bound confirmed superseded by T-190

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

2026-10-12 published T-162 · case 129

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-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

2026-10-13 published T-174 · 11 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-180 · 12 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-182 · case 154

Exact rational certificates at 154, 211 and 241 from Couzo

Upper bound confirmed on these cases, superseded by T-222

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 · cases 129, 154

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 · cases 123, 126, 258

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

2026-10-17 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

2026-10-20 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