T-240: Couzo's 268 and 303 of #763, below T-174 and T-239, from Kaggle notebook packings

The cases

Results on these cases

19 results in the register on $n=268,303$$n=268,303$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=268,303$$n=268,303$

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   S3 V0 C1 Lower bound incomplete on these cases, superseded by T-083 and T-180 (reported)

   Nagamochi · Nagamochi 2005 · source · register

2. 2026-09-04 published T-085 · $n=268,303$$n=268,303$

   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=268,303$$n=268,303$

   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   S3 V3 C3 Upper bound confirmed superseded by T-240 and T-242

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

4. 2026-09-29 published T-083 · $n=268,303$$n=268,303$

   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. 2026-10-03 published T-092 · $n=303$$n=303$

   Smaller packings again at seven counts from $n=208$$n=208$ to $306$$306$, each certified two independent ways

   S3 V3 C3 Upper bound confirmed superseded by T-240

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

6. 2026-10-05 published T-098 · $n=268,303$$n=268,303$

   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   S2 V3 C3 Upper bound confirmed on these cases, superseded by T-240 and T-242

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

7. 2026-10-07 published T-113 · $n=303$$n=303$

   s(n)≤Sn​$s(n)\le S_n$ at ten counts, $n=108$$n=108$ to 303, from new SQUISH rational packings

   S3 V3 C3 Upper bound confirmed superseded by T-240

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

8. 2026-10-09 published T-134 · $n=303$$n=303$

   Six exact rational certificates from Couzo

   S3 V3 C3 Upper bound confirmed on these cases, superseded by T-240

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

9. 2026-10-09 published T-136 · $n=303$$n=303$

   Seventeen exact rational packings from SQUISH's third request

   S3 V3 C3 Upper bound confirmed superseded by T-240

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

10. 2026-10-09 published T-138 · $n=303$$n=303$

    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-240

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

11. 2026-10-10 published T-146 · $n=303$$n=303$

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

    S2 V3 C3 Upper bound confirmed on these cases, superseded by T-240

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

12. 2026-10-12 published T-162 · $n=303$$n=303$

    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-240

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

13. 2026-10-13 published T-174 · $n=268,303$$n=268,303$

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

    S1 V3 C3 Upper bound confirmed on these cases, superseded by T-240 and T-242

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

14. 2026-10-13 published T-180 · $n=268,303$$n=268,303$

    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

15. 2026-10-14 published T-186 · $n=303$$n=303$

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

    S1 V3 C3 Upper bound confirmed on these cases, superseded by T-240

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

16. 2026-10-22 published T-230 · $n=303$$n=303$

    Daniel's smaller n303 packing, started from Ostashev's packing, certified exactly

    S3 V3 C3 Upper bound confirmed superseded by T-240

    Daniel after Ostashev, Levy · Daniel n303 2026-10-22 · packet · register

17. 2026-10-23 published T-239 · $n=303$$n=303$

    Kristiansen's smaller n303 packing, found from Daniel's packing, certified exactly

    S2 V3 C3 Upper bound confirmed superseded by T-240

    Kristiansen after Daniel, Ostashev, Levy · Kristiansen n303 2026-10-23 · packet · register

18. 2026-10-23 published T-240 this result · $n=268,303$$n=268,303$

    Couzo's 268 and 303 of #763, below T-174 and T-239, from Kaggle notebook packings

    S3 V3 C3 Upper bound confirmed

    Couzo after Lavrentev, Tymchishin, Muglich, Polkovskyi, Daniel, Fang, de Winter, Levy · Couzo issue 763 2026-10-23 · packet · register

19. 2026-10-24 published T-242 · $n=268$$n=268$

    Daniel's exact descents of public Kaggle packings at 268 and 304, below the reports before them

    S2 V3 C3 Upper bound confirmed

    Daniel after Lavrentev, Tymchishin · Daniel Kaggle descents 2026-10-24 · packet · register