If there are research models on exactly two floors, then which one of the following statements can be false?

Irina on September 3, 2019

@bb427,

Let's look at the setup for this game.
Cars are displayed on each floor of a 3-floor building. On each floor, the cars are either family or sports, new or used, all production or all research. So on each floor, all the cars have three attributes, and it is one of two choices for each of the attributes.

We can visually set it up in a 3 x 3 grid pattern:

F/S
N/U
P/R
1 2 3

The following rules apply:

(1) If the exhibition includes both family and sports cars, then each family car is displayed on a lower-numbered floor.

F > S

Notice that the condition only applies "if the exhibition includes" both, this tells us that it is not a requirement for the exhibition to include both, it is thus possible to have family cars on all three floors or sports car on all three floors. But if there are both types of cars, we cannot have family cars on floor 3 or sports cars on floor 1 because family cars must always be on lower numbered floors in this scenario.

(2) The exhibition includes no used research model.

This rule tells that if all cars displayed on a particular floor are research models, all cars on this floor are new.

R -> N

If the cars on a particular floor are used, they must be production models:

U -> P

Can we conclude that if the cars are new, they must be research models or that if the cars are production models, they must be used?

No. This would be an invalid inference. It is possible to have new or used production models.

(3) The exhibition includes no research models that are sports cars.

This rule tells us that if sports cars are displayed on a certain floor, all these cars are production models.

S -> P
~ P -> ~ S

If the cars are research models, they must be family cars.

R -> F

We cannot infer though that if a car is a production model, it is a sports car. It is possible to have production models of sports or family cars.

(4)There are new cars on floor 1.
(5) There are used cars on floor 3.


F/S
N/U N U
P/R
1 2 3

Per rule (2), we can conclude that cars on floor 3 must be production models because research models cannot be used.

F/S
N/U N U
P/R P
1 2 3

Now the questions asks us if there are research models on exactly two floors, then which of the following can be false. Can be false question is the opposite of must be true question, so every answer choice except the correct one MUST BE TRUE.

Let's consider this scenario. If there are research models on exactly two floors, and we can see from our initial setup that floor #3 contains production models, we can infer than research models must go on floors 1 &2.


F/S
N/U N U
P/R R R P
1 2 3

Per rule (2), we know that all research models must be new, thus the cars on floor #2 must be new.


F/S
N/U N N U
P/R R R P
1 2 3

Per rule (3), we know that research cars must be family cars, hence we can conclude that floors 1 & 2 contain family cars.


F/S F F ?
N/U N N U
P/R R R P
1 2 3

We could have either family or sports car on floor 3 though since none of the rules apply to used production cars. The only statement that can be false then is (E) as we are not required to have family cars on floor #3.

Does this help?

Let me know if you have any further questions.