#6 The Mutilated Chess Board
The props for this problem are a chessboard and 32 dominoes. Each domino is of such size that it exactly covers two adjacent squares on the board. The 32 dominoes therefore can cover all 64 of the chessboard squares. But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes. Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.
Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered? If so, show how it can be done. If not, prove it impossible.
Created by Martin Gardner
#5 The Prisoner’s Hats
You and nine other individuals have been captured by super intelligent alien overlords. The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative beings. Unfortunately, they’re not sure whether you qualify, to they decide to give you all a test. Through its universal translator, the alien guarding you tells you the following:
“You will be placed in a single-file line facing forward in size order so that each of you can see everyone lined up ahead of you. You will not be able to look behind you or step out of line. Each of you will have either a black or a white hat on your head assigned randomly,
and I won’t tell you how many of each color there are. When I say to begin, each of you must guess the color of your hat starting with the person in the back and moving up the line. And don’t even try saying words other than black or white or signaling some other way, like intonation or volume; you’ll all be eaten immediately. If at least nine of you guess correctly, you’ll all be spared. You have five minutes to discuss and come up with a plan, and then I’ll line you up, assign your hats, and we’ll begin.”
Can you think of a strategy guaranteed to save everyone?
#4 The Sheik’s Camels
An Arab Sheik tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower will win. The brothers, after wandering aimlessly for days, ask a wise man for advice. After hearing the advice, they jump on the camels and race as fast as they can to the city. What does the wise man say?
#3 The Red Eyed Monks Puzzle
There is a monastery of silent monks with no mirrors and one important rule: no red eyes! If a monk discovers he has red eyes he must leave that night. All is well until a visitor says, “at least one of you has red eyes!” What happens next?
#2 The Prisoners’ Boxes Riddle
The director of a prison offers 100 death row prisoners—who are numbered from 1 to 100—a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner’s number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50 drawers in any order. The drawers are closed again afterwards. If during this search, every prisoner finds his number in one of the drawers, all prisoners are pardoned. If just one prisoner does not find his number, all prisoners die. Before the first prisoner enters the room, the prisoners may discuss strategy—but may not communicate once the first prisoner enters to look in the drawers. What is the prisoners’ best strategy?
#1 The Monty Hall Problem
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?