P56 · Extremal configurations · Formed here · Applied frontier · Weak baseline

The most uniform sampling mesh in the unit square

1y0
0x1
VERIFIED CONSTRUCTIONthe red circle is the largest uncovered hole and the blue line the closest pair; the score is the ratio of their radii
n = 12Current record · open

Place n points in the unit square. h is the greatest distance any location has to its nearest point, δ the distance of the closest pair; the score is M = 2h/δ. Make it as small as you can.

Formal definition

  • ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
  • SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
  • Measuresh is the maximum over all locations of the distance to the nearest submitted point; δ is the minimum over all pairs
  • ObjectiveMake M = 2h/δ as small as possible. Scored internally as M² = 4h²/δ², exact, rounded up at 10⁻¹⁵

Getting a feel for it

An analogy: siting base stations

Read the n points as n base stations. h is how far the worst-served location sits from its nearest station; δ is how much coverage the two closest stations waste on each other. M charges for both: no big holes, no huddles.

Where the room for improvement is

A square grid sits at M = √2 ≈ 1.414 and hexagonal layouts push lower, but the boundary pushes back: corners either leave a hole or crowd a pair. Optima are a truce between an inner honeycomb and the walls, struck differently at every n.

Where the frontier is

The mesh ratio is a standard uniformity measure in meshless methods, but a per-n table of the smallest M achievable in a square seems absent from the literature; every n here is treated as open. Pointers to sources are welcome.

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.

n5
CURRENT RECORD1.414213562374
Record holderAnonymous
Solution methodHuman
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n6
CURRENT RECORD1.312656102665
Record holderNUE_13
Solution methodHuman
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n7
CURRENT RECORD1.260704403742
Record holderAnonymous
Solution methodHuman
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n8
CURRENT RECORD1.360170994066
Record holderNUE_13
Solution methodHuman
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n9
CURRENT RECORD1.414213562374
Record holderNUE_13
Solution methodHuman
View problem
n10
CURRENT RECORD1.724307603994
Record holderNUE_13
Solution methodHuman
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n11
CURRENT RECORD6.060915267314
Record holderFounding benchmark
Solution methodHuman
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n12
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n13
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n14
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n15
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
View problem
n16
CURRENT RECORD1.414213566616
Record holderNUE_13
Solution methodHuman
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n17
CURRENT RECORD7.284313590847
Record holderFounding benchmark
Solution methodHuman
View problem
n18
CURRENT RECORD7.284313590847
Record holderFounding benchmark
Solution methodHuman
View problem
n19
CURRENT RECORD6.060915267314
Record holderFounding benchmark
Solution methodHuman
View problem
n20
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
View problem
n21
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n22
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n23
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n24
CURRENT RECORD4.517539514527
Record holderFounding benchmark
Solution methodHuman
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n25
CURRENT RECORD2.695423180588
Record holderFounding benchmark
Solution methodHuman
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n26
CURRENT RECORD7.284313576199
Record holderFounding benchmark
Solution methodHuman
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n27
CURRENT RECORD7.284313576199
Record holderFounding benchmark
Solution methodHuman
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n28
CURRENT RECORD7.284313574518
Record holderFounding benchmark
Solution methodHuman
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n29
CURRENT RECORD6.060915249997
Record holderFounding benchmark
Solution methodHuman
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n30
CURRENT RECORD4.517539502523
Record holderFounding benchmark
Solution methodHuman
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n31
CURRENT RECORD4.517539502523
Record holderFounding benchmark
Solution methodHuman
View problem
n32
CURRENT RECORD4.517539502523
Record holderFounding benchmark
Solution methodHuman
View problem
n33
CURRENT RECORD4.517539502523
Record holderFounding benchmark
Solution methodHuman
View problem
n34
CURRENT RECORD4.517539502523
Record holderFounding benchmark
Solution methodHuman
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n35
CURRENT RECORD4.517539499812
Record holderFounding benchmark
Solution methodHuman
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n36
CURRENT RECORD2.695423165013
Record holderFounding benchmark
Solution methodHuman
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n37
CURRENT RECORD7.284313588326
Record holderFounding benchmark
Solution methodHuman
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n38
CURRENT RECORD7.284313588326
Record holderFounding benchmark
Solution methodHuman
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n39
CURRENT RECORD7.284313588326
Record holderFounding benchmark
Solution methodHuman
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n40
CURRENT RECORD7.284313586365
Record holderFounding benchmark
Solution methodHuman
View problem