Plain language
What this result means
This is the same test as circle packing in the unit square, but with harder walls. The square is replaced by a triangle or a pentagon, so the contact patterns change and the system has to rebuild the search around new constraints instead of reusing one fixed geometry.
- The three clear beats are pentagon n=15, triangle n=27, and triangle n=30.
- Friedman's page prints values with a trailing plus, so hidden digits may exist. A beat only counts when the new layout clears the whole hidden-digit window.
- Pentagon n=21-30 and triangle n=31-34 had no listed value, so those rows are frontier extensions rather than beats.
Visual notes
How to read the result
Result table
Three clear beats and fourteen first-known container values.
| Cell | Baseline | Numaro | Delta | Note |
|---|---|---|---|---|
| cirRpen n=15 | 2.61251 | 2.612714 | +0.00020 | Viquerat June 2026 |
| cirRtri n=27 | 1.76300 | 1.764217 | +0.00122 | Cantrell July 2011 |
| cirRtri n=30 | 1.86315 | 1.865806 | +0.00266 | Viquerat June 2026 |
| cirRpen n=21 | none listed | 3.088531 | first-known | frontier extension |
| cirRpen n=22 | none listed | 3.165663 | first-known | frontier extension |
| cirRpen n=23 | none listed | 3.247349 | first-known | frontier extension |
| cirRpen n=24 | none listed | 3.327581 | first-known | frontier extension |
| cirRpen n=25 | none listed | 3.391645 | first-known | frontier extension |
| cirRpen n=26 | none listed | 3.476635 | first-known | frontier extension |
| cirRpen n=27 | none listed | 3.539869 | first-known | frontier extension |
| cirRpen n=28 | none listed | 3.610265 | first-known | frontier extension |
| cirRpen n=29 | none listed | 3.681347 | first-known | frontier extension |
| cirRpen n=30 | none listed | 3.745880 | first-known | frontier extension |
| cirRtri n=31 | none listed | 1.894825 | first-known | frontier extension |
| cirRtri n=32 | none listed | 1.923736 | first-known | frontier extension |
| cirRtri n=33 | none listed | 1.956110 | first-known | frontier extension |
| cirRtri n=34 | none listed | 1.988375 | first-known | frontier extension |
Method
How it was found
The system searches circle centers inside each container, then recomputes the best radii for those centers. It keeps only layouts whose score survives the final geometry check and the printed-baseline gate.
- Reused the square-packing search loop, but replaced square walls with triangle and pentagon edge constraints.
- Tried many center layouts using multistart search and topology perturbations.
- Recomputed the best feasible radii for promising center layouts instead of trusting the optimizer's radii.
- Counted a beat only when the final value cleared Friedman's full printed window.
Verification
How it was checked
Each layout is checked from raw coordinates. Every circle must stay inside the triangle or pentagon, every pair of circles must stay disjoint, and the final score is the sum of all radii. For the three record claims, the score also has to clear the hidden digits allowed by Friedman's printed plus sign.
Scope
What is not being claimed
Only the three clear beats are record claims. Values that merely match within Friedman's display window are not counted, and the frontier rows are first-known feasible layouts rather than optimality proofs.
References
Baseline sources
Citation
How to cite
Numaro Autoresearch Team. "Maximizing total radius in triangle and pentagon containers." Numaro Research Report NUMARO-2026-005, 2026.
@techreport{numaro2026FriedmanPackingCenter,
title = {Maximizing total radius in triangle and pentagon containers},
author = {Numaro Autoresearch Team},
institution = {Numaro},
number = {NUMARO-2026-005},
year = {2026},
url = {https://numaro.tech/research/friedman-packing-center-2026/}
}