T-125: Complete rational construction reports at 25 counts

2005 published T-007 · 25 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Lower bound incomplete on these cases, superseded by T-073, T-080, T-083, T-090, T-091, T-180 (reported), T-206 and T-216

Nagamochi · Nagamochi 2005 · source · register

2026-09-04 published T-085 · 25 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 70

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Lower bound confirmed superseded by T-091

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

2026-09-27 published T-046 · cases 51, 70, 86, 88

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

Lower bound recorded superseded by T-090 and T-091

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Upper bound confirmed superseded by T-146, T-174 and T-218

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

2026-09-28 published T-048 · case 51

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

Lower bound confirmed on these cases, superseded by T-090

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-070 · cases 51, 70, 86

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

Lower bound confirmed superseded by T-090 and T-091

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 · 5 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 · 25 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 · cases 70, 86, 88

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-090 and T-091

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

2026-10-01 published T-074 · cases 70, 88

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-091

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, 86

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

Lower bound confirmed superseded by T-090 and T-206

Hosono after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-02 · 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-075 · cases 86, 88

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

Lower bound confirmed on these cases, superseded by T-090 and T-091

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-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 · 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-082 · cases 51, 70, 86, 88

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

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

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 · case 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

2026-10-04 published T-090 · cases 51, 84, 86, 88

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 · cases 70, 88

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 · case 84

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

Lower bound confirmed on these cases, superseded by T-206

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

2026-10-05 published T-098 · 7 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 and T-218

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

2026-10-05 published T-101 · 14 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 on these cases, superseded by T-126, T-127, T-134, T-146, T-173, T-174, T-186 and T-187

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

2026-10-07 published T-113 · cases 108, 126, 129, 130

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 and T-186

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 · 6 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 and T-186

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

2026-10-07 published T-116 · 5 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-174 and T-218

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

2026-10-07 published T-117 · case 105

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

Upper bound confirmed superseded by T-174

Rehwaldt after Couzo and earlier contributors · Rehwaldt Couzo refinements 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-129 · cases 105, 130

Three dated Daniel certificate reports with historical source custody

Upper bound recorded superseded by 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 this result · 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 · 8 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 · cases 105, 108, 127, 131

Eight complete rational refinements from Couzo

Upper bound confirmed superseded by T-174

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

2026-10-08 published T-130 · cases 84, 86, 105, 175

Five follow-up rational refinements from Couzo

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

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

2026-10-09 published T-134 · cases 263, 267

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 · cases 131, 153, 236, 263

Seventeen exact rational packings from SQUISH's third request

Upper bound confirmed on these cases, superseded by T-173, T-174 and T-218

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

2026-10-09 published T-138 · 10 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 and T-186

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-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 · case 175

Six more exact rational certificates from Couzo

Upper bound confirmed superseded by T-186

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 · 11 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-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-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-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 published T-158 · case 175

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-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-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-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-173 · case 153

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 · 24 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-179 · case 146

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 · 25 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

2026-10-14 published T-186 · cases 129, 175

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

2026-10-16 published T-193 · 8 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-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-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-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-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

2026-10-22 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