Brain Teasers

Now these you really can't prepare for. You can only hope you are alert enough and have had enough practice to solve these quickly. Here are some practice ones:

  • You have 9 balls and one two-sided balance. One of the 9 balls are lighter than all the rest. In just two weighings, how can you find out which ball is the lightest?
    1. Split up the 9 into groups of 3 balls. Weigh two groups of 3 against each other.
      • If they weigh the same, then you know that the lightest ball is in the other group of three.
      • If they do not weigh the same, take the lighter group of balls.
    2. Take the group of balls you chose and weigh two of those balls against each other.
      • If those two are even, then the ball NOT on the scale is the lightest one.
      • If they are not even, obviously the lighter ball is the lightest ball.
  • You have 9 jars, each containing an unlimited number of marbles. All but one of these jars contain uniform marbles that weigh exactly 1 gram each. The defective jar contains marbles that weigh less at exactly .1 grams. You have a measuring jar which you can put any number of marbles in and a very precise digital scale. In just one weighing, how can you tell which jar contains defective marbles?
    1. Imagine that these jars are labeled 1-9. Then in the measuring jar, put one marble from Jar 1, two marbles from Jar 2, three marbles from Jar 3, etc. When you weight this measuring jar, if the scale reads xx.1, then the defective marbles are from Jar 1. If the scale reads xx.2, then the defective marbles are from Jar 2. If the scale reads xx.3, Jar 3.
  • You have 2 identical glass balls. You have a 100-story building. You are trying to determine at which floor these glass balls break. What is the most efficient way to determine at which floor these glass balls break?
    1. A typical start would be taking the building and flipping it on its side and looking at this problem like a pivot search problem. A strategy to this would be to try segmenting this building to figure out the smallest number of ball drops to figure out at which floor these balls break.
    2. The hard part is that you only have two balls! Which means that you can't just go and do a binary search thing where you keep splitting your search area in half. You have to start from the lowest floor.
    3. So let's say you're going to try this by starting at Floor 25. You drop a ball at that floor. If it doesn't break, you go up to Floor 50. If not, you start at Floor 1. Obviously, if the target floor was 49, you'd have to drop the ball 50 times. This is not efficient.
    4. Try making this increment smaller: try an increment of 10. Start at Floor 10, drop the ball. If it breaks, start at Floor 1 and go up one by one. If not, go to Floor 20. If not Floor 20, go to Floor 30.
    5. Since the worst case scenario is 99, let's see how many tries it would take with an increment of 10. Floor 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 91, 92, 93, 94, 95, 96, 97, 98, and 99. The number of tries is 19, which is infact the most effective and smallest number of tries. The answer here is increment of 10.
  • You have 50 red marbles and 50 blue marbles. Your have two jars. If I am to choose a jar at random, what ratios of marbles would you put in each jar to increase the probability of me choosing a red marble?
    1. A typical start would be trying 25 red and 25 blue in each jar. If this were the case, you would have .5(25/50) + .5(25/50) = 50%.
    2. Let's try increasing this probability. Try putting just 1 red marble in one jar, and all the other 99 marbles in the other. This would give you .5(1) + .5(49/99) = 74.75% This is almost 3/4!
  • The Gasoline Problem. If you have an empty 3 gallon jug, an empty 5 gallon jug, and a full 8 gallon jug of gasoline, how can you measure out exactly 4 gallons of gas using just these jugs?
    1. So let's put this into a table format (This makes for seven swaps and you get two cans of exactly 4 gallons of gasoline):
3 gallon 5 gallon 8 gallon
0 0 8
0 5 3
3 2 3
0 2 6
2 0 6
2 5 1
3 4 1
0 4 4
  • You have 3 identical rectangles. If you arrange these into one rectangle, and if A is the length and B is the width, what is A / B?
    1. Make the width of each of the identical rectangles to be X. Now after looking at the altogether rectangle, you should see that the length is 2X. Therefore, A = 3X and B = 2X. A/B = 3/2
  • Why is a manhole cover round?
    1. Many reasons. It's easier to move a manhole cover by rolling it if it is round is my favorite.
    2. People are typically round around the waist so it's easier to fit into a circular hole.
    3. A round manhole will not fall through its circular hole but a square manhole will fall through if inserted diagonally.
  • Four members of U2 (Bono, the Edge, Larry and Adam) need to get across a narrow bridge to play a concert. Since it's dark, a flashlight is required to cross, but the band has only one flashlight, and only two people can cross the bridge at a time. (This is not to say, of course, that if one of the members of the band has crossed the bridge, he can't come back by himself with the flashlight). Adam takes only a minute to get across, Larry takes two minutes, the Edge takes five minutes, and slowpoke Bono takes ten minutes. A pair can only go as fast as the slowest member. They have 17 minutes to get across. How should they do it?
    1. The key to attacking this question is to understand that Bono and the Edge are major liabilities and must be grouped together. In other words, if you sent them across separately, you'd already be using fifteen minutes.
    2. What does this mean? That Bono and the Edge must go across together. But they can not be the first pair (or one of them will have to transport the flashlight back). Instead, you send Larry and Adam over first, taking two minutes. Adam comes back, taking another minute, for a total of three minutes. Bono and the Edge then go over, taking ten minutes, and bringing the total to 13. Larry comes back, taking another two minutes, for a total of 15. Adam and Larry go back over, bringing the total time to 17 minutes.
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