Commonly asked puzzles

1. A light bulb is hanging in a room. Outside of the room there are three switches, of which only one is connected to the lamp. In the starting situation, all switches are 'off' and the bulb is not lit.
   The Question: If it is allowed to check in the room only once to see if the bulb is lit or not (this is not visible from the outside), how can you determine with which of the three switches the light bulb can be switched on?
   
   To find the correct switch (1, 2, or 3), turn switch 1 to 'on' and leave it like that for a few minutes. After that you turn switch 1 back to 'off', and turn switch 2 to 'on'. Now enter the room. If the light bulb is lit, then you know that switch 2 is connected to it. If the bulb is not lit, then it has to be switch 1 or 3. Now touching for short the light bulb, will give you the answer: if the bulb is still hot, then switch 1 was the correct one; if the bulb is cold, then it has to be switch 3.
2. Tom has three boxes with fruits in his barn: one box with apples, one box with pears, and one box with both apples and pears. The boxes have labels that describe the contents, but none of these labels is on the right box.
   The Question: How can Tom, by taking only one piece of fruit from one box, determine what each of the boxes contains?
   Tom takes a piece of fruit from the box with the labels 'Apples and Pears'. If it is an apple, then the label 'Apples' belong to this box. The box that said 'Apples', then of course shouldn't be labeled 'Apples and Pears', because that would mean that the box with 'Pears' would have been labeled correctly, and this is contradictory to the fact that none of the labels was correct. On the box with the label 'Appels' should be the label 'Pears'. If Tom would have taken a pear, the reasoning would have been in a similar way.
3. You are standing next to a well, and you have two jugs. One jug has a content of 3 liters and the other one has a content of 5 liters.
   The Question: How can you get just 4 liters of water using only these two jugs?
   Solution 1:
   Fill the 5 liter jug. Then fill the 3 liter jug to the top with water from the 5 liter jug. Now you have 2 liters of water in the 5 liter jug. Dump out the 3 liter jug and pour what's in the 5 liter jug into the 3 liter jug. Then refill the 5 liter jug, and fill up the 3 liter jug to the top. Since there were already 2 liters of water in the 3 liter jug, 1 liter is removed from the 5 liter jug, leaving 4 liters of water in the 5 liter jug.
   Solution 2:
   Fill the 3 liter jug and pour it into the 5 liter jug. Then refill the 3 liter jug and fill up the 5 liter jug to the top. Since there were already 3 liters of water in the 5 liter jug, 2 liters of water are removed from the 3 liter jug, leaving 1 liter of water in the 3 liter jug. Then dump out the 5 liter jug and pour what's in the 3 liter jug into the 5 liter jug. Refill the 3 liter jug and pour it into the 5 liter jug. Now you have 4 liters of water in the 5 liter jug.
4. Charles walks over a railway-bridge. At the moment that he is just ten meters away from the middle of the bridge, he hears a train coming from behind. At that moment, the train, which travels at a speed of 90 km/h, is exactly as far away from the bridge as the bridge measures in length. Without hesitation, Charles rushes straight towards the train to get off the bridge. In this way, he misses the train by just four meters! If Charles would, however, have rushed exactly as fast in the other direction, the train would have hit him eight meters before the end of the bridge.
   The Question: What is the length of the railway-bridge?
   Let the length of the bridge be x meters.
   Running towards the train, Charles covers 0.5x-10 meters in the time that the train travels x-4 meters. Running away from the train, Charles covers 0.5x+2 meters in the time that the train travels 2x-8 meters.
   Because their speeds are constant, the following holds:
   (0.5x-10) / (x-4) = (0.5x+2) / (2x-8)
   which can be rewritten to
   0.5x2 - 24x + 88 = 0 Using the abc formula we find that x=44, so the railway-bridge has a length of 44 meters.
5. There is a water-cask with three different water-taps. With the smallest tap the water-cask can be filled in 20 minutes. With middle the tap the water-cask can be filled in 12 minutes. With the largest tap the water-cask can be filled in 5 minutes. The Question: How long does it take to fill the water-cask with the three taps together?
   The smallest tap fills 1/20 water-cask in 1 minute. The middle tap fills 1/12 water-cask in 1 minute. The largest tap fills 1/5 water-cask in 1 minute. Together they fill 1/20 + 1/12 + 1/5 = 1/3 water-cask in 1 minute. Therefore, the whole water-cask is filled in 3 minutes.