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Riley Armstrong throws two darts at a dartboard, aiming for the center (bullseye). The second dart lands farther from the center than the first. If Riley Armstrong now throws another dart at the board, again aiming for the center, what is the probability that this third throw is also worse (i.e., farther from the center) than his first? Assume Riley's skillfulness with the dart is constant.
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Problem 5A (HARD)
The first class to submit the correct solution will receive a special prize
You are given twelve coins. All appear to be identical, as far as you can tell, but you are told that one of them is a counterfeit. All genuine coins have the same mass, while a counterfeit is either lighter or heavier than a genuine coin.
You are allowed to weigh coins using a two-pan equal-arm balance (sometimes called a scale or scales), as shown in figure 1. The balance provides one of three possible indications: the right pan is heavier, or the pans balance, or the left pan is heavier. The balance has sensitivity sufficient for the task. Note that the result is qualitative not quantitative: there is no indication of how much heavier the heavy pan is.
Figure 1: Two-Pan Equal-Arm Balance (a.k.a Scale)
Your mission, should you decide to accept it, is to identify the countefeit coin and tell whether it is “light” or “heavy”, using at most three weighings.
You are not allowed to tamper with or scrutinze the coins, nor gather any information about them except by the three weighings. You may, if you wish, label the coins if that helps you keep track of which is which.
Problem 5B (Not so Hard)
This morning, while I was walking on the beach, I noticed a small coin glistening in the sun. On closer examination I could read the date 126BC. Thinking that this coin would obviously be worth something, I popped in to my local museum to ask their advice. There were very kind to me but informed me that the coin was worthless. Why?
Click HERE to see the solutions to Problem 5A & 5B
Problem #4A & #4B
I have a drawer full of socks. There are 5 socks of each color: red and green, evenly mixed.
In the morning, when it is very dark, I need to pull out a pair of socks of the same color.
How many must I pull out of the drawer to ensure that I have at least one pair of matching socks?
If you are in a dark room with a candle, a wood stove and a gas lamp. You only have one match, so what do you light first?
Click here to see the solution to problems 4A and 4B
There are three switches in a room on the first floor and a lamp with a regular bulb on the third floor.
Each switch is labeled On and Off. One (and only one) of the switches controls the lamp. There is no possible way for you to see whether the light is on or off from the first floor. You are alone in the house.
· You are allowed to turn on and off the switches as often and as long as you'd like.
· However, you can only go upstairs once to check on the status of the lamp.
Explain how to determine which switch controls the light.
for the answer to Problem #3
Three friends bought a large pizza, but weren’t sure how to slice it. They decided to make one cut each across the pizza.
What is the greatest number of pieces they could get with three cuts?
Click here for the solution to problem #2.