Logic Puzzles: Origins, problems, and games GSS October 1, 2012 Peder Thompson Origins of logic & logic puzzles • Prehistoric development of formal logic in China, India, Greece. • Aristotle: looking for relations of dependence which characterize deductions. He distinguished the validity of conclusions drawn from assumptions from the truth of the premises. • Charles Dodgeson (Lewis Carroll): The Game of Logic • Raymond Smullyan • Saul Kripke, 1960s, modal logic Knights and knaves • William and Richard are residents of the island of knights and knaves. • Question: William says “we are both knaves” – Who is what? • Solution: William is a knave: – William can’t be a knight, since then he would be lying, so William is a knave. But that means he is not telling the truth, so they cannot both be knaves, hence Richard is a knight. Syllogisms • Syllogisms: Given a list of premises, what can be deduced from them? – Example: • Major premise: All humans are mortal. • Minor premise: Socrates is human. • Conclusion: Socrates is mortal. – Another example: • Major premise: All horses have hooves. • Minor premise: No humans have hooves. • Conclusion: Some humans are not horses. More on syllogisms • The conclusion only depends on the premises. • Untrue conclusions can be drawn if the syllogism is not used correctly: – For example: • Some animals can fly. • All pigs are animals. • Therefore, some pigs can fly. – (Is there a way we can fix this? By either changing the hypothesis or conclusion?) • Some supposed “syllogisms” are really just wordplay or incorrect usage: • Nobody is perfect. • I am nobody. • Therefore, I am perfect. A (logical?) problem • A man is stranded on an island covered in forest. One day, when the wind is blowing from the west, lightning strikes the west end of the island and sets fire to the forest. The fire is very violent, burning everything in its path, and without intervention the fire will burn the whole island, killing the man in the process. There are cliffs around the island, so he cannot jump off. How can the man survive the fire? (There are no buckets or any other means to put out the fire) A (not-so-good) Solution: • The man picks up a piece of wood and lights it from the fire on the west end of the island. He then quickly carries it near the east end of the island and starts a new fire. The wind will cause that fire to burn out the eastern end and he can then shelter in the burnt area. • (The man survives the fire, but dies of starvation, with all the food in the forest burnt.) That was annoying… An example • A frog is at the bottom of a 30 meter well. Each day he summons enough energy for one 3 meter leap up the well. Exhausted, he then hangs there for the rest of the day. At night, while he is asleep, he slips 2 meters backwards. How many days does it take him to escape from the well? • Note: Assume after the first leap that his hind legs are exactly three meters up the well. His hind legs must clear the well for him to escape. • Solution: 28 days. Each day he makes it up another meter, and then on the twenty seventh day he can leap three meters and climb out. Tic Tac Toe • Can you place six X's on a Tic Tac Toe board without making three-in-a-row in any direction? Try this one: • Can you connect all nine dots with only four straight line segments without losing contact with the paper while drawing? Some unsolved “logic” problems • Grimm’s Conjecture: – Grimm's conjecture states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. • For example, for the range 242 to 250, one can assign distinct primes as follows: • 242: 11 243: 3 244: 61 245: 7 246: 41 247: 13 248: 31 249: 83 250: 5 • ABC Conjecture: – Shinichi Mochizuki of Kyoto University? • Twin Primes Conjecture – The twin primes conjecture states that there are an infinite number of pairs of primes of the form 2n-1, 2n+1. That is, they differ by 2; for example, 41 and 43. Contest time! • Divide into small groups. • These problems all have logical solutions (no “trick questions”) • Group with most points wins. Questions • Question 1: • How many steps are required to break an m x n bar of chocolate into 1 x 1 pieces? You can break an existing piece of chocolate horizontally or vertically. You cannot break two or more pieces at once (so no cutting through stacks). • Question 2: • A man is caught on the King's property. He is brought before the King to be punished. The King says, "You must give me a statement. If it is true, you will killed by lions. If it is false, you will be killed by trampling of wild buffalo.” But in the end, the King has to let the man go. What was the man's statement? • Question 3: • Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe. The canoe can only carry 100kg. Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and Dusty weighs 40 kg, and they have 20kg of supplies. How do they get across? Quick Round • Fastest answer wins: • Question 4: • There are five gears connected in a row, the first one is connected to the second one, the second one is connected to the third one, and so on. If the first gear is rotating clockwise what direction is the fifth gear turning? Tricky? • Question 5: • Two fathers took their sons fishing. Each man and son caught one fish, but when they returned to camp there were only 3 fish. How could this be? (None of the fish were eaten, lost, or thrown back.) • Question 6: • At a restaurant, how could you choose one out of three desserts with equal probability with the help of a coin? • Question 7: • A blind-folded man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles (possibly of different sizes) with each pile having the same number of cards facing up? • Question 8: • What mathematical symbol can be put between 5 and 9, to get a number bigger than 5 and smaller than 9? • Question 9: • What’s wrong with the following “proof”? Question 10: • You are travelling down a country lane to a distant village. You reach a fork in the road and find a pair of identical twin sisters standing there. One standing on the road to village and the other standing on the road to neverland (of course, you don't know or see where each road leads). One of the sisters always tells the truth and the other always lies (of course, you don't know who is lying). Both sisters know where the roads go. If you are allowed to ask only one question to one of the sisters to find the correct road to the village, what is your question? (So explain why your question will make sure to give you the correct road to the village.) That’s all folks! • Thanks!