P51 · Extremal configurations · Classic

Lighting a unit square · n = 2

Place n unit-brightness point lights in the unit square, where the intensity at any point is the sum of 1/distance² to each light, and maximize the intensity at the darkest point of the square.

Instancen = 2

ObjectiveMaximize minimum intensity

Record comparison

minimum intensity · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	24/5	NUE_13 #31	MinMax Arena ↗
Site record	4.799999	Cphysics	Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

ContainerThe unit square [0, 1]², with lower-left corner at (0, 0) and upper-right corner at (1, 1)
SubmissionThe n light positions lights, together with a claimed minimum intensity that the verifier can rigorously verify
ConstraintsAll lights lie inside the square; multiple lights may share a position, in which case their intensities add
Point intensityP = {p₁,…,pₙ} is the multiset of light positions; each unit-brightness light contributes the inverse square of its distance $I_{P} (x) = \sum_{i = 1}^{n} \frac{1}{{‖ x - p_{i} ‖}^{2}}$
Darkest intensityL(P) is the continuous minimum of the intensity over the entire closed square [0, 1]² $L (P) = \min_{x \in {[0, 1]}^{2}} I_{P} (x)$
ObjectiveMaximize L(P) over all valid light configurations P; the submitted intensity must be a lower bound on L(P) that the verifier can rigorously verify $\max_{P} L (P)$

Read the optimality proof ↗

Layout shown: site record

1y0

0x1

VERIFIED CONSTRUCTIONthe darkest point is at least 4.799999

Getting a feel for it

Why it is hard

The darkest point typically occurs at a saddle point between lights or at a corner of the square. Moving lights to brighten one dark spot dims another region. Uniform grids leave the boundary and corners underlit, so optimal layouts place extra weight near the edges and corners.

Known results

Friedman's public table lists the best known configuration and value for each n. NUE_13 #31 proved that the symmetric n = 2 layout's 24/5 is the continuous optimum (its coordinates are irrational, so no nine-decimal grid optimum is claimed); no other n has an optimality proof. The comparison table shows each external value as an interval at its published precision; reaching that interval means matching the published precision rather than proving optimality.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderCphysics

Solution methodHuman

Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

CphysicsHuman
4.430769→4.799999
2026-08-27
SeiaHuman
4.798582→4.430769
2026-08-24

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with lower-left corner at (0, 0) and upper-right corner at (1, 1). Write light coordinates as decimal strings such as "0.25" (at most nine decimal places), and write the claimed minimum intensity with at most six decimal places.

Submit lights and intensity. Write lights as n coordinate pairs of decimal strings, and intensity as the lower bound you claim for the minimum intensity over the square (at most six decimal places). The verifier rigorously checks this bound, so set it slightly below the true minimum. The intensity at any point is the sum of 1/distance² over all lights.

The current leader's answer

{
  "intensity": "4.799999",
  "lights": [
    [
      "0.091751710",
      "0.500000000"
    ],
    [
      "0.908248290",
      "0.500000000"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 2
}

The current leader's answer

{
  "intensity": "4.799999",
  "lights": [
    [
      "0.091751710",
      "0.500000000"
    ],
    [
      "0.908248290",
      "0.500000000"
    ]
  ]
}

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

本题n=3（之后关于这道的讨论都会发在此，也欢迎各位参与）NUE_13#311 replies2d ago 外部记录大幅更新！NUE_13#317 replies9d ago [证明] p51-n2-v1 最优值 24/5：完整严格证明NUE_13#315 replies10d ago 这个外部记录是不是有点水（NUE_13#315 replies30d ago 对称构型的最优性：本题n=2时的最终证明思路与通用思想NUE_13#310 replies39d ago [证明思路 · p51-n2-v1] 本题n=2时证明思路NUE_13#3112 replies44d ago n=10似乎可以参考我这里的摆法hadsajnc#1460 replies58d ago P51 求解日记 2026 9 6hadsajnc#1460 replies58d ago 经过我不懈的努力，终于...hadsajnc#1461 replies58d ago 还在争！！！hadsajnc#1460 replies59d ago