1998 AMC 12 Problem 20

Question

Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that

(a) the numbers are all different,

(b) they sum to

13,

and

(c) they are in increasing order, left to right.

First, Casey looks at the number on the leftmost card and says, “I don’t have enough information to determine the other two numbers.” Then Tracy looks at the number on the rightmost card and says, “I don’t have enough information to determine the other two numbers.” Finally, Stacy looks at the number on the middle card and says, “I don’t have enough information to determine the other two numbers.” Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card?

2

3

4

5

There is not enough information to determine the number.

Answer: C

Concepts:logic ordered triples

Difficulty rating: 2190

Small hint:

List all increasing positive triples with sum

13

Big hint:

Eliminate triples in the order of the three statements, using what each speaker has learned

Solution:

The eight triples are

\begin{gathered} (1,2,10),(1,3,9),\\ (1,4,8),(1,5,7),\\ (2,3,8),(2,4,7),\\ (2,5,6),(3,4,6). \end{gathered}

Casey’s statement excludes

(3,4,6).

With that known, Tracy’s statement excludes rightmost values

10,9,

and

6.

The remaining triples are

(1,4,8),(2,3,8)

and

(1,5,7),(2,4,7).

A middle value

3

or

5

would now identify the triple, so Stacy’s statement forces the middle value

4.

Thus C is correct.