Problems
Monday April 19, 2010
People have been giving me a lot of problems lately. Luckily I've managed to cope, but goodness knows how many more I can take. Probably a lot more.
Geometry Problem
I have a square of size x. Inside the square sits a blue circle of radius x/2, the largest possible circle that sits inside of the square. At the corners, I have four smaller green circles, the largest possible that will fit in the space there. What is the area not covered by a circle (i.e., the red area)?
Timing Problem
I have two lengths of rope. The ropes burn at non-linear rates, but they will each burn entirely in one hour. Using only these ropes and your choice of a tool to ignite them, how do you measure out 45 minutes?
Hat Problem
There's a competition in which you and two friends decide to enter. The prize is $1 million. The rules are as follows:
- Each of you is randomly given a hat which is either red or blue. There are an infinite number of hats so the chance of getting a red or blue hat is exactly 50%.
- You cannot see the hat you are given, but you can see the hats that your two friends have.
- To win $1 million, the three of you must simultaneously call out the colour of the hat on their own head. No other forms of communication (signalling, blinking, pointing, flailing, etc) are permitted.
- You are allowed to pass, however, if all of you pass, you will forfeit. You will also lose if one of you gets it wrong. To pass, you will say "pass" instead of your hat colour.
Now that you know the rules of the competition, you must formulate a strategy that will maximise the chances of you winning the $1 million. You are allowed to discuss this strategy with your friends beforehand. What is the best strategy, and what are your chances of winning?
Posted by Jeremy at 7:28am on Monday April 19, 2010 :: 0 comments :: 