40 matchsticks are arranged to form a 4 by 4 grid of 16 smaller squares.
What is the fewest number of matchsticks that need to be removed so that there are no squares (of any size) remaining?
To solve this problem, we need to determine the number of matchsticks that need to be removed in order to eliminate all squares.
First, let's count the number of squares that can be formed within the 4x4 grid:
- There is 1 square that spans the entire grid.
- There are 4 squares of size 3x3.
- There are 9 squares of size 2x2.
- There are 16 squares of size 1x1.
Therefore, the total number of squares is 1 + 4 + 9 + 16 = 30.
Next, we need to remove matchsticks in such a way that no squares remain. Removing a matchstick will eliminate all squares that contain that matchstick as an edge. To minimize the number of removed matchsticks, we need to find a strategy that removes the maximum number of squares with each matchstick removal.
One possible strategy is to remove the matchsticks along the perimeter of the 4x4 grid. By removing all the matchsticks on the outer edges of the grid, we eliminate all the squares that span across those edges.
There are 4 sides, and each side has 4 matchsticks. So we need to remove 4 * 4 = 16 matchsticks along the perimeter.
Therefore, the fewest number of matchsticks that need to be removed so that there are no squares remaining is 16.