# Difference between revisions of "1991 AHSME Problems/Problem 28"

## Problem

Initially an urn contains 100 white and 100 black marbles. Repeatedly 3 marbles are removed (at random) from the urn and replaced with some marbles from a pile outside the urn as follows: 3 blacks are replaced with 1 black, or 2 blacks and 1 white are replaced with a white and a black, or 1 black and 2 whites are replaced with 2 whites, or 3 whites are replaced with a black and a white. Which of the following could be the contents of the urn after repeated applications of this procedure?

(A) 2 black (B) 2 white (C) 1 black (D) 1 black and 1 white (E) 1 white

## Solution

$\fbox{B}$ The possible operations are $-3B+1B = -2B$, $-2B-W+W+B = -B$, $-B-2W+2W = -B$, or $-3W+B+W = -2W+B$. Notice that the only way the number of whites can change is from $-2W+B$, so it starts at 100 and only ever decreases by 2, so the final number of whites must be even, eliminating $D$ and $E$. Now observe that we can keep repeating operation 1 ($-2B$) until we get to 2 blacks (and 100 whites) left, at which point we can't take out 3 blacks so we can't use this operation any more. We can now use operation 2 to get to 1 black and 100 whites, then operation 3 to get to 0 blacks and 100 whites, and then keep running operation 4 until we get to 2 whites and 49 blacks. Now run operation 2 to give 2 whites and 48 blacks, 2 whites and 47 blacks, and so on, and keep repeating until you reach 2 whites, which is $B.$