import Mathlib.Data.Finset.Card import Mathlib.Order.Bounds.Basic import Mathlib.Tactic /-! # USA Mathematical Olympiad 2001, Problem 1 Each of eight boxes contains six balls. Each ball has been colored with one of n colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer n for which this is possible. -/ namespace Usa2001P1 def possible_num_colors : Set ℕ := { n : ℕ | ∃ f : Fin 8 → Finset (Fin n), (∀ i, (f i).card = 6) ∧ (∀ x y : Fin n, ∀ i j : Fin 8, i ≠ j → x ∈ f i → y ∈ f i → (¬ (x ∈ f j ∧ y ∈ f j))) } /- determine -/ abbrev min_colors : ℕ := sorry theorem usa2001_p1 : IsLeast possible_num_colors min_colors := sorry

This problem does not yet have a complete solution.