Posts Tagged ‘math problems’

A calculator is broken so that the only

April 27, 2013

A calculator is broken so that the only keys that still work are the sin, cos, tan, arcsin, arccos, and arctan buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

I’ve started reading Zeitz’s The Art and Craft of Problem Solving. This one took me about 90 minutes, though as usual, once I had a solution it seemed obvious. Originally from USAMO 1995. Who comes up with these problems? How? You sit down and say, “Okay, it’s time to invent a problem that can be solved with elementary math, but only if you see some diabolical trick”, and then you do what?

New Problems

March 19, 2010

Here are some nice problems I’ve run across recently. They’re from word of mouth, the internet, and books. Only one of them is a “trick” problem. The problems are pretty much unrelated.

Answers here

Do these first two as fast as you can:

  • 999 * 1001
  • A family has five children. What’s the probability that at least three of them are girls?

You have two glass balls and are in a 100-story building. There’s a window on each floor. When you drop the ball out a window, it may or may not break, depending on how high you are. There’s a certain floor that is the transition from not breaking to breaking, so that if you drop it from that floor or above, it will break, and below that floor, it will not. This floor could be any of the floors from one to one hundred with equal probability. What is the fewest number of drops you need to make to be sure you accurately locate the transition floor?

Your friend puts two balls in a jar. Each ball is either red or green, and your friend chooses the color of each ball with a fair coin flip before putting it in. You come up, open the jar, and without looking can smell that there’s at least one red ball in it (but two red balls smell the same as one). What’s the probability that both balls are red?

Same as last problem, but this time you reach in and pull out a red ball. What’s the probability that the remaining ball is also red?

Suppose that there are two barrels, each containing a number of plastic eggs. In both barrels, some eggs are painted blue and the rest are painted red. In the first barrel, 90% of the eggs contain pearls and 20% of the pearl eggs are painted blue. In the second barrel, 45% of the eggs contain pearls and 60% of the empty eggs are painted red. Would you rather have a blue pearl egg from the first or second barrel?

100 prisoners are to be executed, but they are given a chance to save themselves by playing a game. They will all stand in a single file line, so the prisoner in back can see all the other prisoners and the prisoner in front can see no one. The warden will then put a white or black hat on each prisoner’s head, choosing at random as he gets to the prisoner. Then the warden goes to the prisoner at the back of the line (who can see everyone else’s hat, but not his own) and asks him what color his hat is. He can respond only with either “white” or “black”. If he gets it right, he lives. This continues down the line. Each prisoner can hear the responses of all the prisoners who come before him. If the prisoners are allowed to get together and discuss strategy beforehand, how many of the 100 can be saved?

There are eight pitchers of wine, one of which is poisoned. You have some lab rats to test the wine on. If a rat drinks any poison wine, it will die some time within the next 24 hours. How many rats do you need to use to design a test that is certain to discover the poisoned bottle in 24 hours?

Prove that there exist numbers x and y that are both irrational, but x^y is rational.

Suppose you cut a cone out of a sheet of paper. How does the time it takes the cone to fall to the floor when dropped from the ceiling depend on the radius of the cone?

Take a 6*6 chessboard and and 8*8 chessboard. For each, you’re allowed to make one cut through it along the lines between the squares. This will give you four pieces total. How can you make the cuts so those 4 pieces can be rearranged into a 10*10 chessboard? Try the same thing with other Pythagorean triples.

Some Birthday Problems

February 24, 2010

One of the most famous counter-intuitive problems in probability is, “How many people must there be in a room before the probability that at least two of them have the same birthday exceeds one half?”. Most likely, you’ve heard this problem and its solution (which is 23) before. I seem to remember doing it in the fourth grade. But test your intuition against the following:

  1. How many people must there be in a room before the probability of at least one of them having your birthday exceeds one half?
  2. How does the birthday problem change if we’re not looking for two people with the same birthday, but two people with adjacent birthdays?
  3. How does the birthday problem change if we account for the distribution of birthdays not being flat?
  4. How many people must there be in a room before the probability that every day of the year has at least one person with that birthday exceeds one half? (Ignore Feb. 29)
  5. People walk into a room one by one, and continue doing so until there’s a pair of people in the room with the same birthday. What is the expectation value of the number of people who walk into the room? Is it more, less, or the same as the 23 that solve the birthday problem?
  6. How do the answers to these questions (including the original birthday problem) scale with the length of the year? For example, if the year were twice as long (and days were the same length), would the answer to the birthday problem be 46?

There are other obvious questions, such as how many people to have a 1/2 chance of three people with the same birthday, or how many to have two pairs of repeat birthdays, but these seem more tedious than interesting to me. Here is one more, though, which I got from a problem book:

Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a 365-day year. The owners want to maximize the expected total number of man-days worked per year in a factory. How many workers do factories have in Erewhon?