# Perplexing Math

• You throw three darts onto the surface of a globe, each from a randomly chosen direction. What is the probability that all three darts are in one hemisphere?
• You are a bug sitting in one corner of a cubic room. You wish to walk (no flying) to the extreme opposite corner (the one farthest from you). Describe the shortest path that you can walk.
• An Arab sheikh is old and must will his fortune to one of his two sons. He makes a proposition. His two sons will ride their camels in a race, and whichever camel crosses the finish line last will win the fortune for its owner. During the race, the two brothers wander aimlessly for days, neither willing to cross the finish line. In desperation, they ask a wise man for advice. He tells them something; then the brothers leap onto the camels and charge toward the finish line. What did the wise man say?
• There are ten gnomes who have gotten themselves into quite a predicament. They are in the dungeon of a castle of a tyrannical king. Despite the evilness of the king, he has a silver lining in his heart. He has given the gnomes a chance of survival. Here is the offer: The King lines the gnomes up in a single-file row. This means that the tenth gnome sees the back of the person in front of him, and there is no gnome behind the tenth gnome. The ninth gnome has the tenth gnome behind him and the eighth gnome directly in front of him, and so on. Finally, the first gnome has the second gnome directly behind him, but there is no one in front of the first gnome. The king has a large bag full of many black hats and many white hats. There is not necessarily the same number of black hats as white hats. The king randomly reaches into his bag and places a hat on each of the gnomes. This means that the tenth gnome can see everyone’s hat except his own, the ninth gnome can see everyone’s hat except his own and the tenth gnome’s hat, and so on. The first gnome can see no one’s hat. The king then takes out his gun and puts it to the temple of the tenth gnome. The king asks the gnome, “What color is your hat?” If the gnome answers correctly, he lives and gets freed from the dungeon. If he does not, he dies. He continues up the line in this progression. However, before placing the hats on the gnomes, he allows the gnomes to meet as a group and discuss a strategy to save as many of the gnomes as possible. Imagine that you are one of these gnomes. What strategy would you develop? How many gnomes can you guarantee to save?  REMEMBER: When it is your turn to say the color of your hat you must ONLY say “white” or “black.” If you say anything else, the king will shoot you and all of the remaining gnomes.

## 4 thoughts on “Perplexing Math”

1. Srinivas Rangarajan says:

1. The probability is 1.

2 and 3 as given above.

4. The last gnome calls out the color of the hat that is in highest number from among the nine in front (9 means there it has to be so). Say there are 5 white and 4 black, the last calls out “White”. Based on this information, the ninth gnome now knows the color of his hat as he can see the color of the hats of 8 others in front. This way each one can deduce the color of his hat based on the previous calls. A minimum of 9 can be saved this way. If the last gnome was lucky, he could be saved too!

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