1991 AMC 12 Problem 28

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.