How many matchsticks do you need to remove so that no squares of any size remain?
Nine matchsticks are enough, as seen from the solution below.
To see that eight matchsticks are not enough, notice that removing an inner matchstick reduces the number of 1×1 squares at most by 2. Since there are 16 such small squares, in order to get rid of them all, we need to remove only inner matchsticks. However, in this case, the large 4×4 square will remain.