Grades 11-12 Video Solutions 2021-video 2021 11-12/27

Video Transcription

Problem number 27 states, four houses A, B, C, and D have a pet each. There are four pets in all, a dog, a cat, a mouse, and a rabbit.  Of the following statements, exactly one is correct. The pet in house A is not a rabbit. The pet in house B is not a cat. The pet in house C is not a rabbit.  The pet in house D is not a dog. The pet in house A is not a mouse. Which pet lives in house A?  So we know that one of these statements is true. However, we're going to flip this problem around. If we negate each statement, then we can say that exactly one is false,  and all the remaining ones are true. So let's do this. So in order to negate the statements, let's get rid of all the nots, and this is what we get. So of these five statements,  we know that one is false, and the rest are all true. So let's represent each of these statements in an easy-to-understand diagram. So here, a line is going to represent a statement  that a pet lives in one of the labeled houses. Immediately, we see a contradiction. Negated statement A says that Rabbit lives in house A,  while negated statement C says that Rabbit lives in house C. We know one of these has to be wrong. Let's assume negated statement C is wrong, and negated statement A is correct.  This would imply that negated statement E is also wrong, though because the statement says that the pet in house A is a mouse. Only one of these statements can be wrong,  so this means that our initial assumption was wrong, and that negated statement A is actually false. This way, there are no contradictions,  so all the remaining negated statements are true, and from here we can see that it is, in fact, the mouse that lives in house A. So the question asked us,  which pet lives in house A? And the answer is the mouse. Letter C.

Video Summary

The problem involves determining which pet resides in house A based on provided statements about four houses and their pets: a dog, a cat, a mouse, and a rabbit. By negating the statements and determining only one to be false, we identify contradictions in the assumptions about the pets' locations. Initially considering conflicting statements about a rabbit living in house A or C, it's discovered that negating the initial assumption resolves contradictions. Consequently, all other statements hold true, leading to the conclusion that a mouse resides in house A. Thus, the pet in house A is a mouse.