Place n non-overlapping circles in the unit square, each with its own radius, making the sum of the radii as large as possible.
Formal definition
ContainerA square of side 1, origin (0, 0) at the lower-left corner, (1, 1) at the upper-right
SubmissionExactly n circles, each a triple [x, y, r]: a centre plus its own radius
ConstraintsEvery circle lies entirely inside the square; no two overlap in their interiors, tangency allowed; every radius is positive
ObjectiveMake the sum of the radii as large as possible. Radii are nine-decimal numbers, and the sum is an exact integer sum
Getting a feel for it
Where the room for improvement is
Equal circles are one of the worst strategies here: a few large circles pressed into the walls, with small ones tucked into the gaps they leave, beat any uniform arrangement. The trade of one big circle for several small ones replays in every corner, and the optimum spans an order of magnitude in size.
Where the frontier is
This is problem 6.36 of AlphaEvolve's large-scale mathematical discovery runs. On EinsteinArena, n = 26 was pushed to the exact KKT optimum (the first 45 digits of the sum are known); almost no other n has a published value, and every one of them is open.