Proportional cake-cutting with different entitlements

Imagine you and your friend have a cake to split but you want to make sure it's fair! However, you have different ideas on how much of the cake you're each entitled to.

Here's what you can do:

First, decide on a total size for the cake. Then, divide it into equal-sized pieces.

Next, you and your friend will each "claim" a certain number of the pieces based on your entitlements. For example, if you are entitled to 1/3 of the cake and your friend is entitled to 2/3, you would claim one out of every three pieces, while your friend would claim two out of every three.

After you have each chosen your pieces, take turns choosing pieces from the remaining ones until all the pieces are picked. This ensures that everyone gets a proportional share based on their entitlements.

For example, if there are six pieces and you choose one, then your friend will choose two, followed by you choosing another one, and so on until all the pieces are chosen.

With this method, everyone gets a proportional share of the cake and it's still fair, even if people have different entitlements.