Solution: Stones and Marbles

Original problem: here

First, I apologize that the problem was not carefully worded. The problem statement should have been: find all initial conditions where it's always possible for Bob to empty all the three boxes.

There are 3 boxes, and 2010 stones and 2010 marbles are put arbitrarily inside those three boxes.

At each step, Bob is allowed to either:

1. take a stone from one box, a marble from another box, and put them on the third box

2. add or subtract all boxes by the same number of stones

3. add or subtract all boxes by the same number of marbles

Prove or disprove, that by repeating these steps, Bob can empty all three boxes.

Show and Change

Credit for this problem goes to a friend of Boon Leong Ng.

A ticket to a show costs 10 dollars. There are $(m+n)$ (where $m > n$) people waiting on a line. $m$ of them pays with 10 dollar bills and $n$ of them pays with 20 dollar bills. Assuming that the cashier starts out with no money, what is the probability that the cashier never runs out of change? Assuming the probability is uniform among all possible ordering of people.