T-057: $s(211)\le 14.99796070496771500150<15$$s(211)\le 14.99796070496771500150<15$, the first packing of 211 squares below the grid on record

The case

Results on the case

13 results in the register on $n=211$$n=211$, 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   S3 V0 C1 Lower bound incomplete on this case, superseded by T-083 and T-180 (reported)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. 2026-09-16 published T-057 this result

   $s(211)\le 14.99796070496771500150<15$$s(211)\le 14.99796070496771500150<15$, the first packing of 211 squares below the grid on record

   S3 V3 C3 Upper bound confirmed superseded by T-188

   de Winter · de Winter n211 2026-09-16 · 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   S2 V3 C3 Upper bound confirmed on this case, superseded by T-188

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

6. 2026-10-11 published T-149

   New packings of 155, 156 and 211 squares, each with an exact rational certificate

   S3 V3 C3 Upper bound confirmed superseded by T-188

   Ostashev after Nakajima, Levy, Xu · Ostashev records 2026-10-11 · packet · register

7. 2026-10-12 published T-163

   s(n² − n + 1) < n for every n ≥ 15, and n² − n + F(n) squares below side n, from Fang

   S4 V2 C1 Upper bound reviewed superseded by T-188

   Fang after Arslanov, Mustafin, Shangitbayev, Ellsworth, Stead, de Winter, Daniel, Couzo · Fang excess theorem 2026-10-13 · packet · register

8. 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   S1 V3 C3 Upper bound confirmed on this case, superseded by T-188

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

9. 2026-10-13 published T-175

   New packings of 211, 241 and 304 squares, two with merged tilted bands, each certified exactly

   S3 V3 C3 Upper bound confirmed superseded by T-188

   Ostashev after Nakajima, Daniel, de Winter, Couzo, Levy, Xu · Ostashev merged bands 2026-10-13 · packet · register

10. 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

11. 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-188

    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

12. 2026-10-13 published T-182

    Exact rational certificates at 154, 211 and 241 from Couzo

    S3 V3 C3 Upper bound confirmed on this case, superseded by T-188

    Couzo after Daniel, Ostashev, BuddenPing, Fang, Ellsworth, Levy · Couzo certificates 2026-10-13 · packet · register

13. 2026-10-14 published T-188

    A tighter exact certificate for 211 squares from Couzo, of Ostashev's arrangement

    S2 V3 C3 Upper bound confirmed

    Couzo after Ostashev, Daniel, Levy · Couzo verified bounds 2026-10-14 · packet · register