import Std.Data.List.Basic import Mathlib.Data.Finset.Basic import Mathlib.Data.Finset.Card /-! USA Mathematical Olympiad 1999, Problem 1 Some checkers placed on an n × n checkerboard satisfy the following conditions: a. every square that does not contain a checker shares a side with one that does; b. given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least (n²-2)/3 checkers have been placed on the board. -/ namespace Usa1999P1 def checkerboard (n : ℕ) := Fin n × Fin n def adjacent {n : ℕ} : checkerboard n → checkerboard n → Prop | ⟨a1, a2⟩, ⟨b1, b2⟩ => (a1.val = b1.val ∧ a2.val = b2.val + 1) ∨ (a1.val = b1.val ∧ a2.val + 1 = b2.val) ∨ (a2.val = b2.val ∧ a1.val = b1.val + 1) ∨ (a2.val = b2.val ∧ a1.val + 1 = b1.val ) theorem usa1999_p1 (n : ℕ) (c : Finset (checkerboard n)) (ha : ∀ x : checkerboard n, x ∈ c ∨ (∃ y ∈ c, adjacent x y)) (hb : ∀ x ∈ c, ∀ y ∈ c, ∃ p : List (checkerboard n), List.Chain' adjacent p ∧ List.head? p = x ∧ List.getLast? p = y) : n^2 ≤ c.card * 3 + 2 := sorry

This problem does not yet have a complete solution.