T-087: s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

The cases

Results on these cases

12 results in the register on $n=37,61$$n=37,61$, oldest first, each with what it established and how it stands now.

1. 2005 published T-007 · $n=37,61$$n=37,61$

   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   S3 V0 C1 lower bound incomplete on these cases, superseded by T-063 and T-069

   Nagamochi · Nagamochi 2005 · source · register

2. 2018 published T-087 this result · $n=37,61$$n=37,61$

   s(37)≥53​/2+22​−1$s(37)\ge 5\sqrt{3}/2+2\sqrt{2}-1$ and s(61)≥73​/2+22​−1$s(61)\ge 7\sqrt{3}/2+2\sqrt{2}-1$, by optimal piercing

   S3 V3 C1 lower bound reviewed superseded by T-063 and T-069

   Bašić, Slivková · Basic-Slivkova 2018 · source · register

3. 2026-09-04 published T-085 · $n=37,61$$n=37,61$

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

4. 2026-09-27 published T-045 · $n=61$$n=61$

   Rectangle-density lower bounds replayed at 15 counts in $n=18\dots 78$$n=18\dots 78$

   S3 V3 C3 lower bound confirmed superseded by T-063

   wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · packet · packet · source · review · register

5. 2026-09-27 published T-046 · $n=37,61$$n=37,61$

   Rectangle-density lower bounds reported for 48 counts in $n=18\dots 95$$n=18\dots 95$

   S3 V0 C0 lower bound recorded superseded by T-063 and T-069

   wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 rectangle bounds 2026 · wand125 rectangle bounds 2026-09-28 · packet · packet · register

6. 2026-09-28 published T-063 · $n=61$$n=61$

   $s(61)=8$$s(61)=8$, by monotonicity from T-062

   S1 V3 C3 optimality confirmed

   Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

7. 2026-09-29 published T-058 · $n=37,61$$n=37,61$

   Rectangle-certificate ceiling α·UB(n) proved for $n=1$$n=1$..100; B·UB(n) on 64 grid rows

   S2 V3 C3 method limit confirmed

   wand125 after Tokoharu, Daniel · wand125 tools 2026 · packet · register

8. 2026-09-29 published T-083 · $n=37,61$$n=37,61$

   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 on these cases, superseded by T-063 and T-069

   Karakuş · Karakuş 2026 · source · register

9. 2026-09-30 published T-064 · $n=61$$n=61$

   s(k2−3)=k$s(k^2-3)=k$ for every integer $k$$k$ from 6 up; $k=6\dots 18$$k=6\dots 18$ are the cases held here

   S4 V3 C3 optimality confirmed on these cases, second certificate

   Daniel after Burns, Massaccesi · evand square-packing 2026-10-01 · packet · packet · packet · packet · source 1 · source 2 · review 1 · review 2 · register

10. 2026-10-01 published T-069 · $n=37$$n=37$

    Mixed rectangle-measure lower bounds replayed at $n=37,65,66,90,92$$n=37,65,66,90,92$

    S3 V3 C3 lower bound confirmed

    wand125 after Tokoharu, Levy, Stromquist, Nagamochi, Burns, Massaccesi · wand125 mixed bounds 2026-10-01 · packet · source · review · register

11. 2026-10-05 published T-101 · $n=37$$n=37$

    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

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

12. 2026-10-07 published T-124 · $n=61$$n=61$

    Reported non-strict local minima for 178 source configurations

    S2 V0 C0 restricted optimality recorded

    Daniel after Couzo · Daniel exact and local reports 2026 · packet · register