Originally posted by talzamir
so.. if someone moves a bucketful from black to red, and you can only move to black to red to white to black.. is the only way to get the proportions to be the same in a finite time to put all the beans together and then dividing by three?
I haven't done the math but so long as there is a probability you could end up with a particular arrangement, then my guess is that yes it is entirely possible you could end up with the same proportions. The probability will be minuscule but it is possible.
Consider the following example:
Container#1 contains 15 White, Container#2 contains 15 Red & Container#3 contains 15 Black
move 1: move 10W from C#1 to C#2
C#1 contains 5W, C#2 contains 10W & 15R, C#3 contains 15B
move 2: move 10R from C#2 to C#3
C#1 contains 5W, C#2 contains 10W & 5R, C#3 contains 10R & 15B
move 3: move 10B from C#3 to C#1
C#1 contains 5W & 10B, Bag#2 contains 10W & 5R, C#3 contains 10R & 5B
move 4: move 5W & 5B from C#1 to C#2
C#1 contains 5B, C#2 contains 15W & 5R & 5B, C#3 contains 10R & 5B
move 5: move 10W from C#2 to C#3
C#1 contains 5B, C#2 contains 5W & 5R & 5B, C#3 contains 10W & 10R & 5B
move 6: move 5W & 5R from C#3 to C#1
C#1 contains 5W & 5R & 5B, C#2 contains 5W & 5R & 5B, C#3 contains 5W & 5R & 5B
Solved. Absolutely minuscule probability and I'm guessing we need to use a lottery probability calculator for this. So this now comes down to defining a "finite time". If that was defined as while the sun was still burning, no problem. If however, this must be completed before the beans rot, then it is highly unlikely.
Yet another interesting problem.