T-229: Polkovskyi's re-solves of the house packings at 105 and 204, below T-174 and T-225

The cases

Results on these cases

25 results in the register on $n=105,204$$n=105,204$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=105,204$$n=105,204$

   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$

   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-073, T-080, T-083 and T-180 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=105,204$$n=105,204$

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

   S3 V3 C3 Correction confirmed

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

3. 2026-09-27 published T-056 · $n=105$$n=105$

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

   S3 V3 C3 Upper bound confirmed superseded by T-174

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

4. 2026-09-29 published T-083 · $n=105,204$$n=105,204$

   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$

   S3 V3 C3 Lower bound confirmed

   Karakuş · Karakuş 2026 · source · register

5. 2026-10-02 published T-073 · $n=105$$n=105$

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

   S3 V3 C3 Lower bound confirmed

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

6. 2026-10-02 published T-080 · $n=105$$n=105$

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

   S3 V3 C3 Lower bound confirmed

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

7. 2026-10-05 published T-101 · $n=204$$n=204$

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

   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-225

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

8. 2026-10-07 published T-117 · $n=105$$n=105$

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

   S2 V3 C3 Upper bound confirmed superseded by T-174

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

9. 2026-10-07 published T-129 · $n=105$$n=105$

   Three dated Daniel certificate reports with historical source custody

   S1 V0 C0 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

10. 2026-10-08 published T-125 · $n=105$$n=105$

    Complete rational construction reports at 25 counts

    S3 V3 C3 Upper bound confirmed superseded by T-174

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

11. 2026-10-08 published T-128 · $n=105$$n=105$

    Eight complete rational refinements from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

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

12. 2026-10-08 published T-130 · $n=105$$n=105$

    Five follow-up rational refinements from Couzo

    S3 V3 C3 Upper bound confirmed superseded by T-174

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

13. 2026-10-09 published T-138 · $n=105$$n=105$

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

    S3 V3 C3 Upper bound confirmed superseded by T-174

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

14. 2026-10-12 established T-155 · $n=204$$n=204$

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

    S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225

    Levy after Daniel · register

15. 2026-10-12 published T-162 · $n=204$$n=204$

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

    S3 V3 C3 Upper bound confirmed superseded by T-225

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

16. 2026-10-12 published T-170 · $n=105$$n=105$

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

    S1 V3 C3 Upper bound confirmed superseded by T-174

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

17. 2026-10-13 published T-174 · $n=105,204$$n=105,204$

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

    S1 V3 C3 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

18. 2026-10-13 published T-180 · $n=105,204$$n=105,204$

    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

    S4 V0 C1 Lower bound reviewed

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

19. 2026-10-14 published T-186 · $n=204$$n=204$

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

    S1 V3 C3 Upper bound confirmed on these cases, superseded by T-225

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

20. 2026-10-16 published T-193 · $n=204$$n=204$

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

    S2 V3 C3 Upper bound confirmed on these cases, superseded by T-225

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

21. 2026-10-17 published T-226 · $n=204$$n=204$

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

    S2 V0 C0 Upper bound recorded superseded by T-225

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

22. 2026-10-18 published T-223 · $n=204$$n=204$

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

    S3 V3 C3 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

23. 2026-10-18 published T-228 · $n=204$$n=204$

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

    S2 V0 C0 Upper bound recorded superseded by T-225

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

24. 2026-10-20 published T-225 · $n=204$$n=204$

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

    S2 V3 C3 Upper bound confirmed

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

25. 2026-10-22 published T-229 this result · $n=105,204$$n=105,204$

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

    S2 V0 C0 Upper bound recorded

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