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Cheese cutting

This problem comes from our sister site NRICH, which is packed with mathematical problems, games and articles for all ages. The problem is similar to the kind girls will be tackling at the European Girls' Mathematical Olympiad, which will take place in Cambridge next year.

I have a cube of cheese and cut it into pieces using straight cuts from a very sharp cheese wire. In between cuts I do not move the pieces from the original cube shape.

For example, with just one cut I will obviously get two smaller pieces of cheese, with two cuts I can get up to 4 pieces of cheese and with three cuts I can get up to 8 pieces of cheese, as shown in the picture:

cheese cube

Suppose I now make a fourth cut. How many individual pieces of cheese can I make?

Suppose now that I am allowed more generally to cut the block N times. Can you say anything about the maximum or minimum number of pieces of cheese that you will be able to create?

Although you will not be able to determine the theoretical maximum number of pieces of cheese for N cuts, you can always create a systematic cutting system which will generate a pre-detemined number of pieces (for example, making N parallel cuts will always result in N+1 pieces of cheese). Investigate developing better cutting algorithms which will provide larger numbers of pieces. Using your algorithm what is the largest number of pieces of cheese you can make for 10, 50 and 100 cuts?

Here is a hint

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