Math and Puzzles
6.01: Two ropes. You have two ropes, and each takes exactly one hour to burn. How would you use them to time exactly 15 minutes? Note that the ropes are of uneven densities, so half the rope length-wise does not necessarily take half an hour to burn.
Light rope 1 on both ends, when the flames touch, light rope 2. When rope2 = 1/2 lenght, light other end of rope 2.
6.02: Nine Balls. You have nine balls. Eight are of the same weight, and one is heavier. You are given a balance which tells you only whether the left side or the right side is heavier. Find the heavy ball in just two uses of the scale.
Weight 2 groups of 3. Set 1 group of 3 off to the side. Deduce heaviest group from first scaling. Weight last group of 3, weight 2 balls, set one off to the side. Deduce heaviest group.
6.1 The Heavy Pill: You have 20 bottles of pills. 19 bottles have 1.0 gram pills, but one has pills of weight 1.1 grams. Given a scale that provides an exact measurement, how would you find the heavy bottle? You can only use the scale once.
Place n = 1,2,3,... 20 bills respectively from bottles 1,2,3, ... 20 to bottles n+1 = 2,3,4 ... 20. This will give you a hash index into the bottled pill. weight - 210 / .1.
6.2 Basketball: You have a basketball hoop and someone says that you can play one of two games.
Game 1: You get one shot to make the hoop.
Game 2: You get three shots and you have to make two of three shots.
If p is the probability of making a particular shot, for which values of p should you pick one game or the other?
Assuming making a shot is .5, which I will denote as 1. We can have 1 0 0, 0 1 0, or 0 0 1. Which given P(G2) = 3(.5)^3, while P(G1) = .5. So chose G1.
6.3 Dominos: There is an 8x8 chessboard in which two diagonally opposite corners have been cut off. You are given 31 dominos, and a single domino can cover exactly two squares. Can you use the 31 dominos to cover the entire board? Prove your answer (by providing an example or showing why it's impossible).
Not possible. There are 8x8 = 64 - 2 spots. 31 dominos accumpy 31 x 2 = 62 places. But each domino must occupy a white a black square. If for a 64 piece board there are 32 of white and black, for a 62 board there are 30 of one and 32 of the other. Because these are not equivalent and the dominos must occupy both of equivalent amounts, this is not possible.