**The Question:** Suppose is infinitely differentiable and that for each there is some for which the -th derivative of at equals 0. Must be a polynomial?

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**The Question:** Recall that the cyclotomic polynomials are defined for so that . Is it true that every coefficient of every cyclotomic polynomial is either 0, 1, or -1?

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**Background:** Here is a wonderful photograph:

This picture was taken by an astronaut onboard the Apollo 17 spacecraft in 1972. This was the last manned spaceflight to the moon and the last time someone was far enough away from earth to take a picture like this.

**The Question:** When the astronaut took this picture, how far from earth was he?

**Source: **The very nice blog: http://sucolex.blogspot.com/

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**The Warm-up Question: **If every point of the plane is colored either red or blue, do there necessarily exist two points of the same color that are exactly 1 unit apart?

**The Question:** ** **If every point of the plane is colored either red, blue, or green, do there necessarily exist two points of the same color that are exactly 1 unit apart?

**Source:** The main question is part (a) of problem A4 of the 1988 Putnam Exam

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**The Story:** I spent a summer in Chicago and early one morning I watched the sun rise over Lake Michigan (the lake is large enough that you cannot see across it, at least from Chicago). I stood at the waters edge and waited for the first rays of sunlight to peak over the water. When that happened I immediately laid down on the sand near the waters edge, and could no longer see the sun. It took about 10 seconds before the “first” rays of sun were again visible. By the way, when I am standing, my eyes are roughly 5 ft above the ground.

**The Question:** What is the radius of the earth?

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**The Question:** You start with a unit square (say a piece of paper). Choose two adjacent sides of the square and randomly pick a point on each. Connect the two points with a line and cut along that line, thus removing a triangle (it could be a degenerate triangle) from the piece of paper. Let A be the area of the triangle that has been removed.

What is the smallest A can be?

What is the largest A can be?

What is the probability that A is less than 1/4? In particular, is this probability equal to 50%? (after all, 1/4 is half way between 0 and 1/2).

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**The Question:** Suppose you start with an object that has the mass of a proton. If this object doubles in mass every five minutes, how long will take for it to grow to the mass of the current universe?

Ask for guesses. Will it be a few hours? A few years? A few millenia? Much longer?

If you wish, give them the following estimates: Google says the mass of a proton is about kilograms. The wikipedia page on the “Observable Universe” says it has a mass of about kilograms.

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**The Card Trick:**

Ask for a volunteer, who will then pick a card from a standard 52 card deck, show it to the class, and return it to the deck in the location of his/her choice. Shuffle the deck several times and offer the volunteer the opportunity to shuffle if he/she desires. Then randomly choose a card and pronounce with complete confidence, “here is your card.”

Case 1: You guess right and gain the reputation for being a master magician.

Case 2: You guess wrong and get to pose a very nice question. Tell the students: “Someday this is going to work. And it is going to be fantastic!” Then ask: “How many times would I need to perform this trick to have at least a 50% chance of being successfully at least once?”

**Similar Problems:**

1) How many regular 6-sided dice should you roll to have at least a 99% chance of getting at least one 5?

2) How many people must you meet to have a greater than 50% chance that at least one of them shares your birthday?

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