Casework: Start with the Most Limiting Case
Idea · Beginner
Problem Solving
Constraints can be based on the number of elements, on the size of the elements, or on other conditions.
But usually, if you base your answer on the most limiting case, it will make the problem easier to solve.
How many ways are there to color nonadjacent cells in a white row with black?
Base your casework on the middle cell.
Case 1: Middle cell is black
Then the two adjacent cells to that should be white.
For and there are
ways.
Case 2: Middle cell is white
For and there are three ways. Only black-black is not allowed.
Similarly, for and there are three ways.
Therefore,
How many ways are there to color nonadjacent cells in a white table with black?
Base your casework on the middle column.
Case 1: Middle column has one black
WLOG, assume the top cell is black and at the end multiply it by 2.
Case 2: Middle column has two white cells
Therefore,