Counting apples A group of 10 friends stands in a circle. Initially, one person has 200 apples, and
no one else has apples. The friends split the apples according to the following rules:
• sharing: to share, a friend passes two apples to the left and one apple to the right.
• eating: the apples must also be eaten and enjoyed. However, no friend wants to be
selfish and eat too many apples. Every time a person eats an apple, they must also
pass another apple to the right.
A person may only share if they have at least three apples, and they may only eat if they
have at least two apples. The friends continue sharing and eating, until so many apples
have been eaten that no one is able to share or eat any more. How many apples are there which can no longer be eaten?

The key point here is to prove that the final number of apples (in fact, the final configuration of apples) is independent of the sequence of moves used to get there.
Hint: suppose that in a particular sequence of moves you share at some point, and the person to your left's next action is to eat an apple. The combined effect of these two moves is that one of your apples is eaten (it doesn't matter who eats it) and one of them is moved to the person to your right. So this sequence of moves has the same final outcome as some other sequence with fewer moves.
