Unit circles in a minimum-area triangle
Also known ascircles in arbitrary trianglesunit circles in a minimum-area trianglecircle packing in a triangle of variable shape
Place n radius-1 circles with disjoint interiors inside an arbitrary triangle. The triangle's three vertices are part of the answer. Minimize its area.
Formal definition
- ContainerThe container is a non-degenerate triangle whose shape is not fixed in advance; its vertices and all centres lie in the 0-to-200 coordinate frame
- SubmissionThree triangle vertices in counter-clockwise order and exactly n circle centres
- CirclesEvery circle has fixed radius 1; circle interiors are pairwise disjoint and tangency is allowed
- ContainmentEvery circle lies wholly inside the triangle and may be tangent to an edge
- ObjectiveMake the triangle area as small as possible
Getting a feel for it
The container is a variable
Ordinary packing moves the circles; here all three edge directions move too. A tiny edge rotation can release several contacts at once—or destabilize the whole packing.
The hard part is the contact structure
A candidate optimum is usually jammed by circle-circle and circle-edge tangencies. The real search is not uniform spacing but finding the right contact graph among many possibilities.
Starting beyond the trivial cases
Friedman's public table gives the current best-known constructions through n=50. The n=1 and n=2 rows are proved and n=3 is still the elementary equilateral arrangement, so the arena starts at n=4; matching a published construction reaches the frontier, while beating it sets a new record.
SourceONE LEADERBOARD PER n
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓n4
CURRENT RECORD19.392304883194661986best known 38.784609766389323972
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n5
CURRENT RECORD24.124355666451007344best known 48.248711332902014688
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n6
CURRENT RECORD24.124355666451007344best known 48.248711332902014688
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n7
CURRENT RECORD31.212177711947200976best known 62.424355423894401952
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n8
CURRENT RECORD35.9364228891812222best known 71.8728457783624444
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n9
CURRENT RECORD38.784609709944460208best known 77.569219419888920416
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n10
CURRENT RECORD38.784609709944460208best known 77.569219419888920416
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n11
CURRENT RECORD47.003545337013688734best known 94.007090674027377469
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n12
CURRENT RECORD51.304021060395713326best known 102.608042120791426652
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n13
CURRENT RECORD54.894965371157002032best known 109.789930742314004064
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n14
CURRENT RECORD56.908965369437913072best known 113.817930738875826144
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n15
CURRENT RECORD56.908965369437913072best known 113.817930738875826144
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n16
CURRENT RECORD66.033321126116812544best known 132.066642252233625088
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n17
CURRENT RECORD70.275067632551456887best known 140.550135265102913775
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n18
CURRENT RECORD74.803353076891126776best known 149.606706153782253552
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n19
CURRENT RECORD77.346174784918263171best known 154.692349569836526343
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n20
CURRENT RECORD78.497422644931365936best known 156.994845289862731872
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n21
CURRENT RECORD78.497422644931365936best known 156.994845289862731872
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n22
CURRENT RECORD89.12774431764677208best known 178.25548863529354416
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n23
CURRENT RECORD92.741380547858820234best known 185.482761095717640468
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n24
CURRENT RECORD96.497422808550163389best known 192.994845617100326778
Matches the best known
Answer sourceRonald L. Graham and Boris D. Lubachevsky
Solution methodPublished reference construction
n25
CURRENT RECORD99.746134116153205383best known 199.492268232306410766
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n26
CURRENT RECORD102.704871209199083931best known 205.409742418398167862
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n27
CURRENT RECORD103.5499815364248188best known 207.0999630728496376
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n28
CURRENT RECORD103.5499815364248188best known 207.0999630728496376
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n29
CURRENT RECORD115.292243152047020714best known 230.584486304094041428
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n30
CURRENT RECORD118.848622460782341636best known 237.697244921564683273
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n31
CURRENT RECORD122.58804487656187252best known 245.176089753123745041
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n32
CURRENT RECORD127.07490417923377167best known 254.14980835846754334
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n33
CURRENT RECORD129.206351254338314956best known 258.412702508676629912
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n34
CURRENT RECORD131.585923053515497916best known 263.171846107030995832
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n35
CURRENT RECORD132.066642043918271664best known 264.133284087836543328
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n36
CURRENT RECORD132.066642043918271664best known 264.133284087836543328
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n37
CURRENT RECORD144.936031294801118248best known 289.872062589602236496
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n38
CURRENT RECORD148.371411296017511577best known 296.742822592035023154
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n39
CURRENT RECORD151.83073130385406932best known 303.66146260770813864
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n40
CURRENT RECORD155.13843908194960075best known 310.2768781638992015
Matches the best known
Answer sourceRonald L. Graham and Boris D. Lubachevsky
Solution methodPublished reference construction
n41
CURRENT RECORD160.110551326331916238best known 320.221102652663832477
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n42
CURRENT RECORD161.887091638703899028best known 323.774183277407798056
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n43
CURRENT RECORD163.719331331074244432best known 327.438662662148488864
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n44
CURRENT RECORD164.047404167411724528best known 328.094808334823449056
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n45
CURRENT RECORD164.047404167411724528best known 328.094808334823449056
Matches the best known
Answer sourceStandard triangular-lattice construction
Solution methodPublished reference construction
n46
CURRENT RECORD177.887371360134541611best known 355.774742720269083222
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n47
CURRENT RECORD183.094231364901489952best known 366.188462729802979904
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n48
CURRENT RECORD184.75052444774735425best known 369.5010488954947085
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
n49
CURRENT RECORD189.655099647113421046best known 379.310199294226842092
Matches the best known
Answer sourceRonald L. Graham and Boris D. Lubachevsky
Solution methodPublished reference construction
n50
CURRENT RECORD194.498962963929252736best known 388.997925927858505472
Matches the best known
Answer sourceTej Stead
Solution methodPublished reference construction
Data and citation
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/unit-circles-in-minimum-area-triangle.json
The first frozen edition is published at the start of next month; until then, cite with the date you accessed it.
BibTeX (click to copy)
@misc{minmaxarena-unit-circles-in-minimum-area-triangle,
title = {{Unit circles in a minimum-area triangle} (P66)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, accessed 2026-08-29},
url = {https://minmaxarena.com/problems/unit-circles-in-minimum-area-triangle},
license = {CC BY 4.0}
}DISCUSSION
Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
No threads yet. Be the first to talk about this problem?