2012 CEMC Gauss (Grade 8) Problems/Problem 22

Problem

In downtown Gaussville, there are three buildings with different heights: The Euclid (E), The Newton (N) and The Galileo (G). Only one of the statements below is true.

1. The Newton is not the shortest.
2. The Euclid is the tallest.
3. The Galileo is not the tallest.

Ordered from to in height, the buildings are

Solution

We can do multiple cases, where one specific option is true and the others are false. We also have to keep in mind that the buildings have different heights.

Assume that the third statement is true, and the first and second are false. Since "The Newton is not the shortest." is false, The Newton must be the shortest. We then have:

N _ _

Since "The Euclid is the tallest." is false, we know that it must either be the shortest or the second shortest of the three buildings in this case. Since N is already the shortest, it must be the second shortest, giving:

N E _

This leaves G as the tallest building. However, this is a contradiction because "The Galileo is not the tallest." was supposed to be true. Thus, the third statement cannot be true.

Assume that the second statement is true, and the others are false. Since "The Galileo is not the tallest." is false, we know that The Galileo must be the tallest. However, we reach a contradiction already because "The Euclid" was supposed to be tallest because the second statement was true. Thus, the second statement cannot be true.

We can now assume that the first statement is true, and the others are false. Since "The Galileo is not the tallest." is false, The Galileo is the tallest building. We then have:

_ _ G

Because "The Newton is not the shortest." is true, it must either be the tallest or the second tallest building of the three. However, since The Galileo is the tallest, it must be the second tallest. This gives:

_ N G

We then see that the order must be . We can also see that this works with the rules.

~anabel.disher