# New Zealand IMO training 2011 Combinatorics: Cookie Passing

I found this problem:

There are 2n people seated around a circular table, and m cookies are distributed among them. The cookies may be passed around under the following rules:

• Each person may only pass cookies to his or her neighbours.

• Each time someone passes a cookie, he or she must also eat a cookie.

Let A be one of these people. Find the least m such that no matter how m cookies are distributed to begin with, there is a strategy to pass cookies so that A receives at least one cookie.

What I'm looking for is just an example, like a small case verification ( say n = 3 or 4 ), because What I see is that for n=3, taking m=6 is sufficient so that everyone gets a cookie, but they claim that the minimum is $2^n$. Can someone explain where I am wrong ?

• I think the issue is that you cannot pass stacks of cookies – Peter Woolfitt Nov 28 '15 at 20:30
• @PeterWoolfitt: from the answer of "Brazillian Cérebro" , I understand that to pass n cookies you have to eat n cookies. I thought that to pass whatever amount of cookies, you just have to eat one cookie. – Michael Heidelberg Nov 28 '15 at 20:39

Suppose that we have the persons $1,2,3,4,5,6$ sitting next to eachother, with $1$ sitting next to $6$. Suppose person $1$ has six cookies and he wants them to pass to person $4$. He needs three times two cookies to pass three cookies to to person $2$. So now person $6$ doesn't have any cookies and person $2$ now has three cookies. Person $2$ can pass only one cookie to person $3$, and is left with one cookie. So now person $2$ and $3$ both only have one cookie, so they can't pass anything.
Because it doesn't help to pass cookies from person $1$ to $6$ in the beginning, we see that six cookies is not sufficient.