Question

Initially an urn contains $100$ black marbles and $100$ white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows:

Marbles removed	Replaced with
$3$ black	$1$ black
$2$ black, $1$ white	$1$ black, $1$ white
$1$ black, $2$ white	$2$ white
$3$ white	$1$ black, $1$ white

Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure?

$2$ black marbles

$2$ white marbles

$1$ black marble

$1$ black and $1$ white marble

$1$ white marble

Answer: B

Concepts:invariant constructive process parity

Difficulty rating: 2330

Small Hint:

The number of white marbles always changes by $0$ or $2$

Big Hint:

When an operation first leaves at most two marbles, inspect the possible outputs of that final operation

Solution:

The parity of the number of white marbles never changes, so choices with one white marble are impossible. Also a final operation starting with three marbles can leave $1$ black, $1$ black and $1$ white, or $2$ white, but never $2$ black. The state with $2$ white is attainable: repeatedly replace $3$ black by $1$ black until $2$ black remain; twice use the $1$ -black- $2$ -white rule, leaving $100$ white; then alternate the $3$ -white rule with the $1$ -black- $2$ -white rule, reducing the number of white marbles by $2$ per pair until $2$ remain.

Thus the correct answer is B.

1991 AMC 12 Problem 28

Answer: B

Difficulty rating: 2330

Small Hint:

Big Hint:

Solution:

Problem 28 in Other Years