(Can be found in Leino’s puzzle page)

You have 12 coins, 11 of which are the same weight and one counterfeit coin which has a different weight from the others. You have a balance that in each weighing tells you whether the two sides are of equal weight, or which side weighs more. How many weighings do you need to determine: which is the counterfeit coin, and whether it weighs more or less than the other coins. How?

## Archive for December 2007

### The fake coin

December 15, 2007### The electrician problem

December 15, 2007(Can be found in Leino’s puzzle page)

You’re an electrician working at a mountain. There are N wires running from one side of the mountain to the other. The problem is that the wires are not labeled, so you just see N wire ends on each side of the mountain. Your job is to match these ends (say, by labeling the two ends of each wire in the same way).

In order to figure out the matching, you can twist together wire ends, thus electrically connecting the wires. You can twist as many wire ends as you want, into as many clusters as you want, at the side of the mountain where you happen to be at the time. You can also untwist the wire ends at the side of the mountain where you’re at. You are equipped with an Ohm meter, which lets you test the connectivity of any pair of wires. (Actually, it’s an abstract Ohm meter, in that it only tells you whether or not two things are connected, not the exact resistance.)

You are not charged for twisting, untwisting, and using the Ohm meter. You are only charged for each helicopter ride you make from one side of the mountain to the other. What is the best way to match the wires? (Oh, N>2, for there is no solution when N=2.)

### Fair division among N people

December 15, 2007(Given by Mahdi)

You have some amount of gold dust, and you want to share it among N people such that every person agrees to have exactly 1/N of the gold. The main problem is that there is no measurement cup to help dividing.

This problem is well known for 2 people: one divides the gold and the second chooses a half. How can this be generalized for N people?

### Flipping cards

December 15, 2007(Can be found in Leino’s puzzle page)

You’re given a regular deck of 52 playing cards. In the pile you’re given, 13 cards face up and the rest face down. You are to separate the given cards into two piles, such that the number of face-up cards in each pile is the same. In separating the cards, you’re allowed to flip cards over. The catch: you have to do this in a dark room where you cannot determine whether a card is face up or face down.

### The breaking crystal balls

December 15, 2007(Given by Tom)

You have two equal crystal balls for which we know that when dropped from a building with 100 floors will break. You just don’t know which is the first floor from where the balls will break.

To find out the floor you can go to the first floor, drop one of the balls, and if it doesn’t break you go to the second floor and try again, until the limit is found. Having 2 balls, how can you minimize the maximal number of tries until the limit is found?

### Burning ropes to measure time

December 15, 2007(Can be found in Leino’s puzzle page)

You are given a box of matches and a piece of rope. The rope burns at the rate of one rope per hour, but it may not burn uniformly. For example, if you light the rope at one end, it will take exactly 60 minutes before the entire rope has burnt up, but it may be that the first 1/10 of the rope takes 50 minutes to burn and that the remaining 9/10 of the rope takes only 10 minutes to burn. How can you measure a period of exactly 45 minutes? You can choose the starting time. More precisely, given the matches and the rope, you are to say the words “start” and “done” exactly 45 minutes apart.

### Find the light bulb

December 15, 2007(Given by Dave)

In the ground floor you have three switches, each for a lamp in the first floor. Initially all the switches are turned off, but you don’t know which switch corresponds to which light bulb. How can you find the correct matching between switches and light bulbs by changing the switches and go upstairs only once?