Suleyman Cenk Sahinalp presents:

Brain Bogglers

(E) elementary, my dear Watson.

(T) busywork.

(A) simple but needs understanding.

(H) needs both skill and patience.

(T) You are at the crossing of the roads to heaven and hell where you see two demons. One of the demons (you don't know whom) always tells the truth, while the other always lies. You are to find the road to heaven with ONE question, asked to one of the demons. What is the question?

(E) You go one mile to south, then one mile to west, and finally one mile to west to find out that you reached your starting location. Where are you? (There is a less obvious answer to this question.)

(T) You have two jars, each capable of holding 10 liters of any liquid. One of the jars holds 6 liters of water, while the other holds 6 liters of %100 alcohol. Find a method to have 6 liters of %50water-%50alcohol solution in each jar. (Assume that there is no volume loss when alcahol and water is solved in each other. Using another cup is not permitted.)

(A) You have a rectangular shaped table and unlimited quarters to play a game with your opponent. One move of the game is putting one quarter on the initially empty table. The game lasts until a quarter can not be placed on the table. The last to put a quarter is to win. You can not put quarters on top of each other. Suppose you start first. How can you beat your opponent?

(T) You have twelve identical looking balls among which all but one of them are of equal weight. You do not have any information about the weight of the balls. Find the ball with different weight by using a balance just three times.

(E) You are given ten boxes each but one having ten golden coins. The coins in one of the boxes are made of silver. Knowing that a gold coin weighs 50 grams and a silver coin weighs 25 grams, find which box has silver coins after ONE weight measurment.

(E) Three viziers of the grand sultan are blindfolded and told that either a red or a green fez will be put on the head of each. The blidfolds are then removed so that each of the tree can see others' fezes but not his. Actually all tree viziers are given red fezes. The sultan asks to raise their hand whenever they see a red fez, so all three raise their hands. Then he are asks to lower their hands if they know what they are wearing. The most clever among them does.. How?

(E) A bloody tzar wants to kill an imprisoned rebel leader, yet he wants to be seen as a fair ruler by his people. He thinks for a while to come up with the following 'trick': In the presence of a huge crowd, he offers the rebel leader to choose one of the two folded pieces of paper on which he claims to have written the words 'death' and 'life'. In reality both pieces of paper have the word 'death' on them. Knowing the tzar is the stupidest and yet the cruelest creature among human kind, what is the best strategy for the rebel leader to stay alive?

(E) You are given an 8X8 board divided into 64 squares and two diagonally opposite corners are removed. You are also given a pile of dominoes and each domino can cover two squares. Pave the surface of the board with the dominoes. Each domino must lie on two squares.

(H) A question from ancient egypt: Which set of integers 'a' satisfy the equality, 1/a = 1/b + 1/c, for only two pairs of integers 'b' and 'c'?

(E) Grand caliph Harun Rashid of Baghdad calls forty of his best man and says: "Some of your wives are cheating on you. Kill your wife in the end of the day that you understand she is cheating!". It happens that all the wives of these men are cheating on them. It is known that both the men and women of Baghdad are very clever and very talkative. The men like to talk about their affairs with women to everybody except their husbands, and the wives are very successful in making their husbands not to suspect them. Hence all men know that 39 of the wives are cheating but none of them is suspicious about his own wife. 39 nights pass without any occasion. In the end of the fortieth night all men kill their wives. Why?

(T) Prove that the geometric object, which has the maximum area among all the geometric objects with the property of having the distance between any two of their points being upper bounded by a constant 'c', is the circle with radius '2c'.

(H) In how many diffenet ways can you divide a polygon into non-overlapping triangles?

(T) Prove that in a group of six people, either there is a group of three who know each other or a group of three who do not know each other.

Suleyman Cenk Sahinalp (jenk@umiacs.umd.edu)