Question 4
Place $k$ marbles in the cells of a $2024\times2024$ square board so that each cell contains at most one marble and no two marbles occupy adjacent cells. Two cells are adjacent if they share a side.
(a) For $k=2024$, give such a placement in which moving any marble to any adjacent cell produces a placement no longer satisfying both conditions.
(b) Find the largest $k$ such that, for every placement satisfying the two conditions, some marble can be moved to an adjacent cell so that no two marbles are adjacent in the new placement.
(a) For $k=2024$, give such a placement in which moving any marble to any adjacent cell produces a placement no longer satisfying both conditions.
(b) Find the largest $k$ such that, for every placement satisfying the two conditions, some marble can be moved to an adjacent cell so that no two marbles are adjacent in the new placement.