T-166: A tighter rational realization of PingYou Ltd's packing of 153 squares

The case

Results on the case

14 results in the register on $n=153$$n=153$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)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. 2026-09-29 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

4. 2026-10-05 published T-101

   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   S2 V3 C3 Upper bound confirmed on this case, superseded by T-173

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

5. 2026-10-07 published T-114

   $s(153)\le 7250614903299225/562949953421312$, from a SQUISH rational packing

   S3 V3 C3 Upper bound confirmed superseded by T-173

   Chaoweeraprasit after Ellsworth · SQUISH n153 2026-10-07 · packet · packet · register

6. 2026-10-08 published T-125

   Complete rational construction reports at 25 counts

   S3 V3 C3 Upper bound confirmed superseded by T-173

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

7. 2026-10-08 published T-127

   Fourteen rational refinements, seventeen complete source cases

   S3 V3 C3 Upper bound confirmed on this case, superseded by T-173

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

8. 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   S3 V3 C3 Upper bound confirmed on this case, superseded by T-173

   Chaoweeraprasit after Couzo, Xu, Göbel, Schadt, Ellsworth, Hajba, Levy · SQUISH third request 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-173

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

10. 2026-10-11 published T-148

    A packing of 153 squares built from Xu's packing of 70, its contacts reclosed at a new angle

    S3 V3 C3 Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n153 certificate 2026-10-11 · packet · register

11. 2026-10-12 published T-166 this result

    A tighter rational realization of PingYou Ltd's packing of 153 squares

    S1 V3 C3 Upper bound confirmed superseded by T-173

    PingYou Ltd after Xu, Daniel, Levy · PingYou n153 refinement 2026-10-12 · packet · register

12. 2026-10-12 published T-173

    Polkovskyi's packings of 153 and 207 squares, each below the earlier refinements there

    S2 V3 C3 Upper bound confirmed

    Polkovskyi after Xu, Chaoweeraprasit, Levy · Mishapolk exact certificates 2026-10-12 · packet · register

13. 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, pending adoption

    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

    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