Question 4
Two types of trees are planted on an $m\times n$ rectangular grid, one tree per cell. A planting is called impressive if:
i) The total numbers of the two types are equal.
ii) In every row the difference between the numbers of the two types is at least half the number of cells in that row, and in every column the analogous difference is at least half the number of cells in that column.
a) Exhibit an impressive planting for $m=n=2016$.
b) Prove that the existence of an impressive planting implies that both $m$ and $n$ are multiples of $4$.
i) The total numbers of the two types are equal.
ii) In every row the difference between the numbers of the two types is at least half the number of cells in that row, and in every column the analogous difference is at least half the number of cells in that column.
a) Exhibit an impressive planting for $m=n=2016$.
b) Prove that the existence of an impressive planting implies that both $m$ and $n$ are multiples of $4$.