Flipping Coins

Someone walks into your room and dumps a huge bag of quarters all over the floor. They spread them out so no quarters are on top of any other quarters. a robot then comes into the room and is programmed such that if it sees a head, it flips it to tails. If it sees a tail, it throws it in the air. the robot moves around randomly forever. Will there be a convergence in distribution of heads vs. tails?

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Pennies

I challenge you to a game. we each get one penny and we flip them at the same time. (so on turn 1, we each flip our respective pennies – turn 2, we flip them again, and so on until someone wins). I am looking to get heads then tails. You are looking to get heads then heads. So if you flip heads on any flip and then heads on the next flip, you win. If I flip heads on any flip and then tails on the next flip, I win. (its not a speed race, we both flip at the same time, except i’m only concerned with what appears on my coin, and you are only concerned with whats on your coin). Are the odds fair? (obviously not, otherwise this wouldn’t be a question). who has the advantage and why?

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Last Ball

You have 20 blue balls and 14 red balls in a bag. you put your hand in and remove 2 at a time. If they’re of the same color, you add a blue ball to the bag. If they’re of different colors, you add a red ball to the bag. (assume you have a big supply of blue & red balls for this purpose. note: when you take the two balls out, you don’t put them back in, so the number of balls in the bag keeps decreasing). What will be the color of the last ball left in the bag?

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Coin Rolls

Every night, I dump all the change in my pocket into a big bucket.

When I buy things, I never hand over coins. always bills. So I accumulate a lot of coins. Even if the purchase price is $1.01, and I have lots of coins in my pocket, I pay $2 and take the 99 cents in change. All the more coins to dump in my change bucket!

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Calendar Cubes

A man has two cubes on his desk. every day he arranges both cubes so that the front faces show the current day of the month. what numbers are on the faces of the cubes to allow this?

Solution

First, to show all possible days, we’d need one of each of the ten digits. We’d also need two 1s and two 2s to show 11 and 22. That’s twelve numbers right there. Two cubes, twelve faces, so every face is used. Quite elegant.

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Orbs

You have two identical crystal orbs. you need to figure out how high an orb can fall from a 100 story building before it breaks. you know nothing about the toughness of the orbs: they may be very fragile and break when dropped from the first floor, or they may be so tough that dropping them from the 100th floor doesn’t even harm them.

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