thomson-100

Metric coulomb-energy, minimize · Best known 4448.35 (Wikipedia Thomson table) · Play · Code · All example problems

hillclimb problem get thomson-100      # copies the problem into hillclimb/problems/
hillclimb verify thomson-100           # scores the floor; spread 0, the verifier is exact
hillclimb run thomson-100 --budget 15m

The problem

Place exactly 100 unit point charges on the unit sphere so that their Coulomb energy U = sum over the C(100,2) = 4950 pairs of 1 / |p_i - p_j| (units e = 1, k_e = 1) is as small as possible.

Constraints (verified programmatically):

• exactly 100 rows with finite coordinates
• every row is a non-zero vector; the verifier normalizes each row to unit length, so you may submit any non-zero direction
• no two charges may coincide (that makes the energy infinite)

This is a classic hard continuous optimization problem. The best known energy for n = 100 is 4448.350634 (Wikipedia Thomson table). The landscape has many local minima that differ only in the 3rd–4th decimal, so a precise local optimizer matters as much as the start. Good approaches: a Fibonacci-spiral or random start, then gradient descent / L-BFGS with the gradient projected onto the tangent plane and the points renormalized after every step, basin hopping or simulated annealing over the local minima, and restarts that keep the best. numpy and scipy are available.

Submission format

Write submission.csv in the working directory with the header id,x,y,z and 100 rows (id = 0..99), like sample_submission.csv (a weak valid baseline: points on a few latitude rings).

Scoring

The orchestrator runs problem/verify.py after your script finishes. The verifier normalizes the rows, computes the Coulomb energy and prints val_score: <energy> (100000.0 if invalid). Lower is better.

There is no train/test data; this is a pure optimization problem. Keep total runtime well within the execution time limit.