challenges / erdos-squares-in-square
Erdős squares in a square, maximum total side length
Erdős' 1932 problem (AlphaEvolve repository problem 55): place n squares of any sizes and orientations in the unit square, interiors disjoint, to maximise the sum of their side lengths, for n in {10, 12, 14, 17, 26, 37, 50}. Scored against the conjecturally optimal k + c/k constructions.
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# Erdős squares in a square, maximum total side length
## Goal
`pack.py` exposes `pack(n: int, time_budget: float, seed: int) -> list[tuple[float, float, float, float]]`:
`n` squares `(cx, cy, theta, s)` inside the unit square `[0, 1] x [0, 1]`, where `(cx, cy)` is the
centre, `theta` the rotation in **radians** (0 means axis-aligned; any real value is accepted) and
`s >= 0` the side length. Interiors must be pairwise disjoint (touching is fine). Maximise `sum(s)`.
Let `f(n)` be the maximum. Erdős asked in 1932 whether `f(k^2 + 1) = k`, i.e. whether one extra
square buys nothing over the trivial `k x k` grid. Erdős–Soifer (1995) and Campbell–Staton (2005)
independently gave the construction `f(k^2 + 2c + 1) >= k + c/k` for `-k < c < k` and conjectured
it is optimal; Praton (2005, arXiv:math/0504341) showed that conjecture is equivalent to the
original one. Baek, Koizumi and Ueoro (2024, arXiv:2411.07274) proved it when all squares are
…