Question

- Some people stand in a circle exchanging pieces of chocolate. Each of them starts with an even number of chocolates (the numbers might be different between different people). Every minute, each of the people passes half their chocolates to the person to their right. If anyone ends up with an odd number of chocolates, they take an extra chocolate from a jar in the center. Prove that regardless of the initial distribution of chocolates, after a finite number of steps everyone ends up with an equal number of chocolates​

Answer 1

Answer:

If the people started off with a even amount of chocolates, then it is most likely for half of them to also be an even number too.

Step-by-step explanation:

Answer 2

This proposition has been proven; as follows:

Explanation:

• Given

1. people stand in a circle exchanging pieces of chocolate
2. everyone starts with an even number of chocolates
3. Every minute, each of the people passes half their chocolates to the person to their right
4. If anyone ends up with an odd number of chocolates, they take a bar of extra chocolate from a jar in the centre.

• To Prove -- regardless of the initial distribution of chocolates, after a finite number of steps everyone ends up with an equal number of chocolates​
• Assumptions

1. Let 2M be the max no. of chocolates in any player's hand
2. let m be the minimal number
3. let r be the number of players with 2m chocolates

• PROOF:

1. If a player has 2a pieces and their left neighbour has 2b pieces, in the next round, this player will have $(2a-a)+b=a+b\leq 2M$ pieces.
2. Even if a+b is odd, it is $\leq 2M-1$ and so after refill from the jar, it is still $\leq 2M$
3. Likewise, the new amount is ≥2m with equality if and only if a=b=m.
4. We conclude that a situation (M,m,r) turns into (X,x,f) with m≤x≤X≤M. Moreover, if x=m, then f≤r.
5. But it cannot happen that (X,x,f)=(M,m,r) unless M=m.
6. if m<M, then there must be some person with 2m pieces while their left neighbour has more. Then this person will have >2m pieces in the next round, meaning that either x>m or at least f<r; both of which are non-real solutions
7. if M>m, then it takes only finitely many steps until M−m decreases by at least one.
8. after finitely many steps, we reach M=m