Logic puzzles who is lying

These problems will illustrate some of the logical concepts we've looked at, as well as illustrating some proof techniques that we'll look at in more detail later. These proofs are written entirely in words, so for the moment we don't need to worry about the presentation details associated with mathematical symbols. The general setup: You're on an island where each inhabitant is a truth-teller or a liar. Truth-tellers always tell the truth; liars always lie. You're given some information about some people, usually in the form of statements they make.

You're asked to determine whether each person is a truth-teller or a liar. In some cases, it may be impossible to determine what everyone is, or the situation may be impossible. You're on an island where each inhabitant is a truth-teller or a liar. Calvin and Phoebe are on the island. In this problem, I notice that I can determine the truth or falsity of the statement "34 is odd" without knowing anything about Phoebe or Calvin.

So I'll start with it and see what follows from it. Since "34 is odd" is false, the "if" part of Phoebe's statement is false. Hence, Phoebe's statement is true. Therefore, Phoebe must be a truth-teller.

Now Calvin says "Phoebe is a liar", and that is false, since I just showed that Phoebe is a truth-teller. Therefore, Calvin is a liar. Thus, Phoebe is a truth-teller and Calvin is a liar. Here are some general ideas about how you might approach a problem like this.

Pick a character in the problem let's say you pick "Leopold". Consider the two cases: "Leopold is a truth-teller" or "Leopold is a liar". You can picture the reasoning process as a tree :. You can start with either case let's suppose you start with "Leopold is a truth-teller". But we can lie, if we want, and every once in a while we do. Well, there is a special place, a faraway island, where there are two large families quite different from us. And the other family always lies —always , even when they wish they could tell the truth!

In fact, they can be trusted more than we can be. But these special families, the Liars and Truthtellers can be trusted always to lie or always to tell the truth, depending on their family. You meet two people, Adam and Alec. What about Alec? If students have no idea, suggest that they just guess about the speaker.

Student: I think Adam is a Liar. If Adam is lying, then… What? Teacher: Right. Student : Well, if he lied, then he has to be a Liar. Teacher: And do we know anything about Alec? Student: I think Adam is a Truthteller. If Adam is telling the truth, then… What? Student: Wait! People — adults and students — who are new to these puzzles often get stumped at this point. Especially if they are not confident about their reasoning, they may attribute the contradiction to bad reasoning.

But instead of recognizing the correctness of their thinking, people who are not used to this kind of problem can, at this point, feel confused. Teacher: Right! People in the Liar family always tell lies. Never, not even by accident, do they ever tell the truth. You get to decide who goes first. Do you have a winning strategy? There is a straight line of N foxholes, and one fox. Every night, the fox moves from his current foxhole to the one either immediately to his left, or immediately to his right.

Every day, you get to look in one fox hole. Give a strategy to guarantee finding the fox in the fewest number of days. There is an island with red and blue eyed monks.

There are no reflections, and nobody speaks about eye color. If they find out their eyes are red, they kill themselves at midnight that night. You go to the island and see that at least one has red eyes. You say "At least one of you has red eyes. If so, when?

There are three people who want to have a duel. You are person C. You get to shoot first. What should you do in order to maximize the chance that you will live, assuming that A and B are logical and also want to maximize their chances of survival? Hint: this would not be here if the answer was completely trivial. You have two lengths of fuse. Each burns for an hour, exactly. They do not burn at a steady rate, so if you cut one in half, then you do not know if a half will burn for 1 second or 59 minutes though the sum of the times of the two halves is one hour.

How do you time exactly 45 minutes? You have a cow in a circular pen radius 1. You want it to only eat half of the grass in the pen. Assume the cow is a point. You tether the cow to the edge of the pen circular enclosure. How long should its tether be so that it can eat exactly half the grass? There is a prison with inmates. The warden strikes a deal with them.

There is a room with a light in it controlled by a light switch. Each day, the warden will take a random prisoner to that room.

At some point, a prisoner must say "We have all been in the room! If he or she is wrong, then all will never be set free. The initial state of the light is not known on or off. Talking is allowed ahead of time, but not after the process begins. Also, the process will begin on a random day, so you do not know if you are the first in or not.

You are a prisoner. What plan do you propose in order to ensure that you will gain your freedom? Find any solution that works - do not worry about how long it will take. The prisoners will be taken in randomly: over an infinite amount of time, they will all enter the room an infinite number of times. Bonus: how long will it take for you to be freed?

You have a very big urn, and pebbles numbered with the natural numbers 1, 2, At time step 1, you put pebbles in the urn. At time step 2, you take out pebble 1. At time step 3 you put in At time step 4 you take out pebble 2, etc. If you did this an infinite number of times, how many pebbles would be left in the urn? Hint: The limit as the number of iterations goes to infinity may give a different answer.

Try to think about this one without using math at first. There is a rubber band attached to a wall. The rubber band is one meter long, levitating horizontally away from the wall. When I say "go", it will start stretching so the end moves at 10 meters per second. It stretches infinitely. The stretch is uniform e. There is an ant starting where the rubber band connects to the wall. It walks at. Will the ant ever reach the end of the rubber band? If so, how long will it take? What if the rubber band doubles in length every second?

A woman and her husband attended a party with four other couples. As is normal at parties, handshaking took place. Of course, no one shook their own hand or the hand of the person they came with. And not everyone shook everyone else's hand. But when the woman asked the other 9 people present how many different people's hands they had shaken they all gave a different answer.

Question this is NOT a trick! You have two very resilient dinosaur eggs. They will absorb a certain amount of force with no negative consequences, but at some point they will crack. If they don't crack, no damage is incurred. You're on a story building. You have 20 trials you're allowed at most 20 individual egg drops and 2 eggs. Is it possible to devise a testing strategy that guarantees to tell you at exactly what floor the eggs will break?

If after your first trial dropping an egg off of the balcony on one floor of the building , if the egg does not break, then you have 19 trials remaining and two eggs. If the egg breaks, then you still have 19 trials remaining, but only one egg left. How few trials do you need? Let this number be k. Using k trials, can you solve a story building?

How about ? You work with Jane. You know that she has two children. One day you meet one at a boys' camp, and it is a boy. What is the probability that she has two boys? Where on the Earth can you walk a mile south, a mile west, and a mile north, and end up exactly where you started? Hint: There are infinite places: find them all!

What is a shape defined by a mathematical equation that has infinite surface area, but finite volume? There are two empty bags.