T-164: A packing of 302 squares from four boundary patches of Xu's packing of 70 and two upright banks

The case

Results on the case

15 results in the register on $n=302$$n=302$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007

   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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085

   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

   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   S3 V3 C3 Upper bound confirmed superseded by T-186

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

4. 2026-09-29 published T-083

   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. 2026-10-05 published T-098

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

   S2 V3 C3 Upper bound confirmed on this case, superseded by T-186

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

6. 2026-10-07 published T-116

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

   S3 V3 C3 Upper bound confirmed superseded by T-186

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

7. 2026-10-09 published T-136

   Seventeen exact rational packings from SQUISH's third request

   S3 V3 C3 Upper bound confirmed on this case, superseded by T-186

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

8. 2026-10-09 published T-138

   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   S3 V3 C3 Upper bound confirmed superseded by T-186

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

9. 2026-10-10 published T-146

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

   S2 V3 C3 Upper bound confirmed on this case, superseded by T-186

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

10. 2026-10-12 published T-164 this result

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

    S3 V3 C3 Upper bound confirmed superseded by T-186

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

11. 2026-10-13 published T-174

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

    S1 V3 C3 Upper bound confirmed on this case, superseded by T-186

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

12. 2026-10-13 published 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

    S4 V0 C1 Lower bound reviewed

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

13. 2026-10-13 published T-181

    Exact rational certificates at 211, 241 and 302 from Couzo

    S3 V3 C3 Upper bound confirmed superseded by 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

14. 2026-10-14 published T-186

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

    S1 V3 C3 Upper bound confirmed

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

15. 2026-10-16 published T-193

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

    S2 V3 C3 Upper bound confirmed

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