A Collection of Brain Teasers...

Here's to the indomitable spirit of puzzle-solvers!! Don't ask for answers. Only hints. You can check your solutions with me once you feel satisfied. If you feel really lost, I can goad you in the right direction. But no answers!

  1. Prove that at any party, there are at least two people who have shaken hands with an equal number of people.

    Thanks to Marlon - after a game of the Settlers of Cratan



  2. There is this weird man who goes to a noodle shop and orders noodle soup. He asks for exactly 100 noodles in his soup.

    Once his noodle-soup arrives, he gets to work. He extracts two noodle ends from under the soup, ties a knot, and lets it slip into the soup again. He does the above 100 times - so that now all the noodle ends are tied.

    Now he reaches his hand into the soup. What is the probability that he would extract a garland containing all the 100 noodles?

    Was it Nagi or Dada who gave me this?



  3. A chess board has 64 squares. Pick numbers between 1 and 64 and place these numbers on each of the squares of the chess board such that the value in every square is the average of the values of its neighboring squares.

    Prove that the only way to make such an arrangement is when all squares have equal numbers.

    Thanks to Venki - apparently this is what he asks those who want to join Goldman for a job.



  4. In a school common room there are 100 locker boxes. At tiffin, 100 students line up outside the common room. One by one, they come in. The 1st student opens every locker box. The 2nd student closes every even locker box. The 3rd student changes the state (opens if closed, closes if open) of every 3rd locker box. And so on. The nth student changes the state of every nth locker box.

    After the 100 students have done their job, which are the locker boxes that remain open?

    Once again, thanks to Marlon - after a game of the Settlers of Cratan



  5. A pack of cards has 52 cards. You are in a dark room with this pack of cards. You have been told that inside the pack there are 42 cards facing down, 10 cards facing up. You have been asked to reorganize this pack of cards into two decks - so that each deck contains an equal number of cards that face up.

    Remember, you are in the darkness and can't see. How will you do it?

    Hint: The two decks need not contain an equal number of cards.

    Np gave me this puzzle over the phone



  6. A Boeing 737-100 has exactly 100 seats for passengers. A 100 passengers are waiting in a line outside this aircraft, their boarding cards in their hands. Now the first passenger gets onto the plane. He drops his boarding card on purpose and chooses a seat at random. When the second passenger comes in, if he sees that his seat is occupied, he chooses another seat at random. If his seat is not occupied, he takes his own seat. Like this, one by one, 99 passengers board the plane. If one finds one's seat occupied, he chooses a seat at random.

    Finally, the 100th passenger enters the plane. There is only one seat left in the plane - he has to take it. What is the probability that he gets the seat assigned to him on his boarding card?

    From Car Talk



  7. There is an evil king. And like all evil characters in James Bond movies, this evil king likes making extremely devious plots - but unfortunately they are all ridden with loopholes. ... Just like in James Bond.

    So anyway, let us come to the point. The king has 20 political prisoners. He wants to kill all of them.

    One night he goes to the prison and tells them that the next morning he is going to make them stand in line and place hats on their heads. White or black hats. They will not be able to see their own hats - but of course, they will be able to see all hats of people in front of them. They will have to guess (in any order - front to back, back to front - as they wish) the color of their own hats. Those who are correct will be pardoned and set free. Those who make a mistake - die.

    That night the prisoners discuss a strategy among themselves.

    The next morning during the hat trial, only one of them becomes a martyr. The remaining 19 are pardoned - as promised by the king - since they could guess their hat's color correctly.

    What strategy did they adopt? What was the great loophole in the king's evil plot?

    Np gave me this puzzle over the phone



  8. This is ancient lore - from the time Empire State Building was being built. They had raised the structure successfully, had made all the floors, had built the elevator shaft. They had laid cable inside the elevator shaft. However, just before installing the elevator, they found out the blunder - the cables were not color coded.

    They contracted this one electrician and asked him to color code all the wires.

    There was a set of N=25 wires running from the top of Empire State Building to the bottom through a cable sheath. The electrician can assume that the wires are all insulated, and the cable sheath is conductive (although this is not necessary for a solution). The only equipment that the electrician has is a continuity meter. He knows how to connect and disconnect wires and how to use the continuity meter.

    Professor Tsividis gave this problem in one of our lab group meetings.



  9. Given two points, without using any ruler or scale, using only a compass, you have to construct the remaining two points of a square, so that the given two points form the vertices of a side of the square.

    Professor Tsividis gave this problem in one more of our lab group meetings.



  10. There is this job for which there are three applicants - there is only one vacancy. The job requires the person to be very intelligent. Somehow all three applicants went through all the sieves equally well. Finally the interviewer devised this ingenious test: He made all three of them sit together in a room. Four hats were brought in the room - three were black and one was yellow. The applicants were blind-folded and then one of the hats was kept aside while the remaining three hats were placed on the heads of the applicants. If the blind folds are opened the applicants would not be able to see their own hats, but they would be able to see the hats on others. Three assistants now stood ready to open the blindfolds. The firt person who would be able to say what the colour of his hat was would get the job.

    At this juncture, before the blinds were opened, one of the participants raised his hand and said that his hat was coloured...

    What was the colour of his hat, and how did he deduce that?

    This was borrowed from Car Talk on WNYC!

  11. This brain teaser is the one I love the most:

    There are these five robbers - let's call them A, B, C, D and E. They rob a bank of 1 million $s. Now they want to divide the money among themselves. They come up with this brilliant plan of division of money:

    First robber A will suggest a method of division of money. All the robbers, including A, will vote on this. If there is an absolute majority for A, the method as suggested will be followed. If A loses in the vote, he will be killed and the money will be divided among B, C, D and E with B suggesting a method of division of money. The same procedure will follow. Note that B needs a vote division of 3-1 for him to get an absolute majority.

    The robbers are all intelligent and have a strict sense of priority in making their decisions. The following are what they want to achieve, in strict sense of priority:

    1. Life
    2. As much money as possible
    3. Kill as many other robbers as possible

    Now it is A's turn to suggest a method of division of money. What will he suggest?
    (Assume that the smallest unit of money is $1.)

    Hint: try out the two robber case first, then go up to three robbers - extend to five robbers.

    I first got this problem from a mail sent by Satish Verma.

  12. This is another classic:

    There is this small island where the people are all very very intelligent, and very very patriotic. However, they cannot speak, read or write. The only mode of communication they have is by looking at each other. They understand English when it is spoken.

    This doctor reaches this island and observes that several islanders have a terrible disease which is very contagious. The only escape from the disease is through death. The doctor calls all the islanders together at their community center and tells them about this. Unfortunately one who has the disease will not know that he has the disease. However others will be able to recognise him as being infected. The doctor tells them that to save the race, all those who have the disease have to commit suicide. The doctor suggests that the islanders meet every day at 7 am in the morning at their community center and look at each other.

    After 10 days all those who had the disease committed suicide on the same day. How many islanders had the disease and how did they find out?

    I was first asked this problem as a youngster by Rajada.



  13. A cheeky problem:

    There is one bulb inside a room but three switches outside the room. Only one of these switches operates the bulb. There is no way to see the state of the bulb from outside the room.

    You have to fiddle with the switches, enter the room once and only once, and have to be able to tell which switch operated the bulb. What will you do?

    If you have answered the first question, extend the problem to four switches.

    Ajith Kamath first asked me this when I was in IITM.



  14. Okay, this is comparatively easy. Fill in the blanks:

    This sentence contains __ 1's, __ 2's, __ 3's, __ 4's, __ 5's, __ 6's, __ 7's, __ 8's, __ 9's and __ 0's.

    There are two possible solutions. Try to find out both of them.

    Shyam asked me this before an Asha meeting.



  15. There is a certain kind of rope which takes exactly one hour to burn from end to end. The rope is not uniform, not symmetric - thicknesses throughout its length vary randomly. No two such ropes will be identical.

    You have two such ropes. How will you measure 45 minutes?

    You have only one such rope. How will you measure 15 minutes?

    Ranjeet asked me these questions at his farewell party.