MinMax Arena
中/EN
P82 · Packing and covering · Classic

Equal circles in regular polygons

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION
m = 5 · n = 6Current record · open

Pack n equal circles into the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1), maximising their radius.

Formal definition

  • ContainerThe editor and certificate normalize to the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCircle radius r, larger is better; the container circumradius is 1.

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
ONE 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) ↓
Instancem = 5 · n = 1
CURRENT RECORD(1+√5)/4
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 5 · n = 2
CURRENT RECORD(1+√5)√(10+2√5)/(4(1+√5+√(10+2√5)))
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 5 · n = 3
CURRENT RECORD0.389795792best known 0.389795792433
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 4
CURRENT RECORD0.354680462best known 0.354680462504
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 5
CURRENT RECORD0.340440645best known 0.340440645254
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 6
CURRENT RECORD0.309016994best known 0.309016994375
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 7
CURRENT RECORD0.277511094best known 0.277511095117
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 8
CURRENT RECORD0.257541824best known 0.257541824500
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 9
CURRENT RECORD0.243640224best known 0.243640224399
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 10
CURRENT RECORD0.230721507best known 0.230721507966
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 11
CURRENT RECORD0.224333592best known 0.224333593340
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 12
CURRENT RECORD0.21738464best known 0.217384640521
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 13
CURRENT RECORD0.207999994best known 0.207999994722
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 14
CURRENT RECORD0.202386815best known 0.202386816023
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 15
CURRENT RECORD0.197735767best known 0.197735768366
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 16
CURRENT RECORD0.191785608best known 0.191785609135
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 17
CURRENT RECORD0.183126736best known 0.183126736787
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 18
CURRENT RECORD0.178473321best known 0.178473321196
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 19
CURRENT RECORD0.173441903best known 0.173441904330
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 20
CURRENT RECORD0.169279411best known 0.169279411749
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 21
CURRENT RECORD0.165404018best known 0.165404018640
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 22
CURRENT RECORD0.161673301best known 0.161673301786
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 23
CURRENT RECORD0.158923366best known 0.158923367389
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 24
CURRENT RECORD0.156027297best known 0.156027298450
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 25
CURRENT RECORD0.15312221best known 0.153122210789
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 26
CURRENT RECORD0.150683336best known 0.150683336700
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 27
CURRENT RECORD0.148137584best known 0.148137584795
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 28
CURRENT RECORD0.145444423best known 0.145444423784
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 29
CURRENT RECORD0.143333012best known 0.143333013135
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 30
CURRENT RECORD0.141698173best known 0.141698173815
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 31
CURRENT RECORD0.138616361best known 0.138616362210
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 32
CURRENT RECORD0.13579361best known 0.135793611068
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 33
CURRENT RECORD0.134134799best known 0.134134799538
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 34
CURRENT RECORD0.131960534best known 0.131960534658
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 5 · n = 35
CURRENT RECORD0.130170592best known 0.130170592942
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 36
CURRENT RECORD0.128215137best known 0.128215138245
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 37
CURRENT RECORD0.126679668best known 0.126679668746
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 38
CURRENT RECORD0.125056797best known 0.125056798044
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 39
CURRENT RECORD0.12418574best known 0.124185741467
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 5 · n = 40
CURRENT RECORD0.122413791best known 0.122413791780
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 6 · n = 1
CURRENT RECORD√3/2
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 6 · n = 2
CURRENT RECORD2√3−3
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 6 · n = 3
CURRENT RECORD0.433012701best known 0.433012701892
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 4
CURRENT RECORD0.375015221best known 0.375015221341
Hard
Answer sourceShahriar Manzoor
Solution methodPublished reference construction
Instancem = 6 · n = 5
CURRENT RECORD0.33340497best known 0.333404971008
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 6
CURRENT RECORD0.316987297best known 0.316987298108
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 7
CURRENT RECORD0.316987297best known 0.316987298108
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 8
CURRENT RECORD0.26958537best known 0.269585370786
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 9
CURRENT RECORD0.249261972best known 0.249261972966
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 10
CURRENT RECORD0.242756322best known 0.242756322541
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 11
CURRENT RECORD0.232050807best known 0.232050807569
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 12
CURRENT RECORD0.232050807best known 0.232050807569
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 13
CURRENT RECORD0.21650635best known 0.216506350946
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 6 · n = 14
CURRENT RECORD0.214290684best known 0.214290684433
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 15
CURRENT RECORD0.20155921best known 0.201559210543
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 6 · n = 16
CURRENT RECORD0.200025786best known 0.200025787346
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 17
CURRENT RECORD0.194813312best known 0.194813312769
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 6 · n = 18
CURRENT RECORD0.19399769best known 0.193997690565
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 19
CURRENT RECORD0.19399769best known 0.193997690565
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 20
CURRENT RECORD0.17615287best known 0.176152871147
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 21
CURRENT RECORD0.175149749best known 0.175149750125
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 22
CURRENT RECORD0.168017815best known 0.168017815756
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 23
CURRENT RECORD0.166338331best known 0.166338331558
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 24
CURRENT RECORD0.163462238best known 0.163462238508
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 25
CURRENT RECORD0.159724476best known 0.159724477011
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 26
CURRENT RECORD0.158493648best known 0.158493649054
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 27
CURRENT RECORD0.158493648best known 0.158493649054
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 28
CURRENT RECORD0.15166001best known 0.151660010858
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 29
CURRENT RECORD0.150002434best known 0.150002435355
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 30
CURRENT RECORD0.150002434best known 0.150002435355
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 31
CURRENT RECORD0.144498617best known 0.144498618555
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 32
CURRENT RECORD0.142897922best known 0.142897923362
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 33
CURRENT RECORD0.142870298best known 0.142870299182
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 34
CURRENT RECORD0.140313381best known 0.140313381692
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 6 · n = 35
CURRENT RECORD0.139827762best known 0.139827762740
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 36
CURRENT RECORD0.139768253best known 0.139768253701
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 37
CURRENT RECORD0.139768253best known 0.139768253701
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 38
CURRENT RECORD0.131624251best known 0.131624252390
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 39
CURRENT RECORD0.129882646best known 0.129882647187
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 6 · n = 40
CURRENT RECORD0.129711778best known 0.129711778826
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 1
CURRENT RECORD√(2+√2)/2
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 8 · n = 2
CURRENT RECORD√(2+√2)/(2+√(2+√2))
Proven optimum
Answer sourcePackomania (author unspecified)
Solution methodPublished reference construction
Instancem = 8 · n = 3
CURRENT RECORD0.435387433best known 0.435387434219
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 4
CURRENT RECORD0.400543816best known 0.400543816310
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 5
CURRENT RECORD0.348203325best known 0.348203325454
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 6
CURRENT RECORD0.317727657best known 0.317727658062
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 7
CURRENT RECORD0.310945216best known 0.310945216831
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 8
CURRENT RECORD0.285852433best known 0.285852433677
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 9
CURRENT RECORD0.27059805best known 0.270598050073
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 10
CURRENT RECORD0.251932403best known 0.251932403490
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 11
CURRENT RECORD0.239556873best known 0.239556874251
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 12
CURRENT RECORD0.234100964best known 0.234100964947
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 13
CURRENT RECORD0.223537234best known 0.223537235211
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 14
CURRENT RECORD0.219348902best known 0.219348902644
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 15
CURRENT RECORD0.21023689best known 0.210236890635
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 16
CURRENT RECORD0.2042866best known 0.204286601358
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 17
CURRENT RECORD0.198194124best known 0.198194125496
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 18
CURRENT RECORD0.193696359best known 0.193696359493
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 19
CURRENT RECORD0.191508243best known 0.191508243904
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 20
CURRENT RECORD0.185530744best known 0.185530744908
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 21
CURRENT RECORD0.180670259best known 0.180670260259
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 22
CURRENT RECORD0.17550094best known 0.175500940767
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 23
CURRENT RECORD0.172561559best known 0.172561559662
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 24
CURRENT RECORD0.168652863best known 0.168652864098
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 25
CURRENT RECORD0.16645089best known 0.166450891366
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 26
CURRENT RECORD0.163606135best known 0.163606136259
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 27
CURRENT RECORD0.161395047best known 0.161395047685
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 28
CURRENT RECORD0.160404294best known 0.160404294689
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 29
CURRENT RECORD0.154929848best known 0.154929849312
Hard
Answer sourceStefan Zorcic
Solution methodPublished reference construction
Instancem = 8 · n = 30
CURRENT RECORD0.152319578best known 0.152319579409
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 31
CURRENT RECORD0.149878152best known 0.149878153406
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 32
CURRENT RECORD0.147282442best known 0.147282442434
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 33
CURRENT RECORD0.145306599best known 0.145306599845
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 34
CURRENT RECORD0.143477531best known 0.143477532364
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 35
CURRENT RECORD0.142256368best known 0.142256369270
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 36
CURRENT RECORD0.139934446best known 0.139934447382
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 37
CURRENT RECORD0.138229583best known 0.138229583693
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 38
CURRENT RECORD0.13609798best known 0.136097981333
Hard
Answer sourcePaolo Amore
Solution methodPublished reference construction
Instancem = 8 · n = 39
CURRENT RECORD0.134961309best known 0.134961309874
Hard
Answer sourceEckard Specht
Solution methodPublished reference construction
Instancem = 8 · n = 40
CURRENT RECORD0.132682453best known 0.132682454149
Hard
Answer sourcePaolo Amore
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/circles-in-regular-polygons.json

Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.

GET https://minmaxarena.com/data/editions/2026-08/circles-in-regular-polygons.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-regular-polygons-2026-08,
  title  = {{Equal circles in regular polygons} (P82)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-08 edition},
  url    = {https://minmaxarena.com/data/editions/2026-08/circles-in-regular-polygons.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

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