a) Determine how much area is removed in steps P1, P2, P3, Pn

b) By considering a suitable series, show that the area of the Sierpinski carpet is equal to 0.

c) Describe a point on the Sierpinski carpet that will never be removed by the algorithm.

In order to solve this problem, you have to analyse what is happening in each square when goes from step 1 to 2,3,4, etc. Look beyond what your eyes can see, be creative. Make assumptions, e.g. from step 1 to 2 think about in how many parts you can divide this square in order to remove just the center of the circle. From step 2 to 3 the same but now how many squares are surrounded to the big white square; note how the number of squares are increasing while you go from step 1 to 3.

As you can see, this exercise looks like a series… Which one? Consider your first term and the ratio (the number of squares that increase) solve by this method, do not forget that the length of the square is 1. Once that you have the result of the sum of the series, you can subtract from the length and you have proved that the area is equal to 0.

Finally, you can conclude that in the square is a part that will never be remove, think about which part you never take in account for solving this problem.