There are 10 more truth-tellers than liars in a room. Truth-tellers always tell the truth, and liars always lie. Everyone in the room was asked, Are you a truth-teller? And everyone gave an answer. A total of 20 people answered, Yes. How many liars were in the room? So in this problem, let's think about what each type of person would say in response to the question. Let's say a truth-teller was asked the question, Are you a truth-teller? Well, truth-tellers always tell the truth, so they would truthfully answer yes, because they are a truth-teller. Now let's think about what a liar would say. Well, a liar is not a truth-teller, but they would lie. And so even though they aren't truth-tellers, they would still lie, and they would answer yes. So that means that since 20 people answered yes, there are 20 people in the room, because every single person in the room would have answered yes to that question. Now let's try and figure out how many liars there are. We need to use the first part of the problem, which is that there are 10 more truth-tellers than liars in a room. So let's draw out the room. You can see that there are 20 people in here, and the people with happy faces are the truth-tellers, and the people with sad faces are the liars. You can see that the value in the cloud is unknown, and that's what we're trying to figure out. But you can see that there are also 10 extra truth-tellers. Since there are 20 people in total, let's draw out this red lasso. This red region contains 10 people inside of it, because we know that we're excluding the 10 extra truth-tellers. In this red region, we know that the number of truth-tellers and the number of liars is the same, because the number of truth-tellers is exactly 10 more than the number of liars, and we've already taken those extra 10 out. This means that these group of 10 people have to be split into two groups of 5 people, and that means that there are 5 truth-tellers and 5 liars among these 10 people. So back to the original problem, if there are 15 truth-tellers and 5 liars, you can check all the conditions of the problem, and all of them will hold. There are 10 more truth-tellers than liars, because 15 is 10 more than 5. The problem asks us how many liars are in the room, and the answer is B, 5.

truth-tellers

liars

logic puzzle

problem-solving

room scenario