Smallest ratio of largest to smallest distance
1y0
123456789101112
0x1
Place n points in the plane; with D the largest and d the smallest pairwise distance, make the ratio D/d as small as possible.
Formal definition
- ContainerNo real container: the ratio is invariant under scaling and translation, so the answer is written scaled into the unit square
- SubmissionExactly n points, no two coinciding
- ObjectiveMake D/d as small as possible; compared internally by the squared ratio, exactly
Getting a feel for it
Where the room for improvement is
The nearest pair must not be near and the farthest must not be far, so the set is forced round and even: hexagonal inside, but won or lost at the boundary.
Where the frontier is
The family is collected on Friedman's maxmin page (constructions by Rechenberg, Cantrell, Audet and others), none proven; three values are recorded here, the rest are open.
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.
n9
CURRENT RECORD2.683281573
123456789
Record holderNUE_13
Solution methodHuman
n10
CURRENT RECORD2.952567023
12345678910
Record holderNUE_13
Solution methodHuman
n11
CURRENT RECORD3.605551276
1234567891011
Record holderFounding benchmark
Solution methodHuman
n12
CURRENT RECORD2.909312918best known 2.909312912
123456789101112
Record holderFounding benchmark
Solution methodHuman
n13
CURRENT RECORD4.242640688
12345678910111213
Record holderFounding benchmark
Solution methodHuman
n14
CURRENT RECORD4.242640688
1234567891011121314
Record holderFounding benchmark
Solution methodHuman
n15
CURRENT RECORD4.242640688
123456789101112131415
Record holderFounding benchmark
Solution methodHuman
n16
CURRENT RECORD3.590157373best known 3.590157366
12345678910111213141516
Record holderFounding benchmark
Solution methodHuman
n17
CURRENT RECORD5
1234567891011121314151617
Record holderFounding benchmark
Solution methodHuman
n18
CURRENT RECORD5
123456789101112131415161718
Record holderFounding benchmark
Solution methodHuman
n19
CURRENT RECORD3.863703315best known 3.863703306
12345678910111213141516171819
Record holderFounding benchmark
Solution methodHuman
n20
CURRENT RECORD5
1234567891011121314151617181920
Record holderFounding benchmark
Solution methodHuman
n21
CURRENT RECORD5.65685425
123456789101112131415161718192021
Record holderFounding benchmark
Solution methodHuman
n22
CURRENT RECORD5.65685425
12345678910111213141516171819202122
Record holderFounding benchmark
Solution methodHuman
n23
CURRENT RECORD5.65685425
1234567891011121314151617181920212223
Record holderFounding benchmark
Solution methodHuman
n24
CURRENT RECORD5.65685425best known 4.574735319
123456789101112131415161718192021222324
Record holderFounding benchmark
Solution methodHuman
n25
CURRENT RECORD5.65685425
12345678910111213141516171819202122232425
Record holderFounding benchmark
Solution methodHuman
n26
CURRENT RECORD6.403124238
1234567891011121314151617181920212223242526
Record holderFounding benchmark
Solution methodHuman
n27
CURRENT RECORD6.403124238
123456789101112131415161718192021222324252627
Record holderFounding benchmark
Solution methodHuman
n28
CURRENT RECORD6.403124238
12345678910111213141516171819202122232425262728
Record holderFounding benchmark
Solution methodHuman