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### Simple but remarkable math facts

I'm putting together a list of simple but remarkable math facts that could be used to spice up a math class, more-or-less at the level of high school math. These should be presentable in the form of questions that could be asked of a class, using some showmanship to draw students into them. I'll give a starter list of questions. Most are from geometry, a couple from probability. Happy to hear from anyone of similar questions.

(1) A length of railroad track is two miles long. To account for expansion in hot weather, the track has been joined in the middle. In hot weather the track expands, with each one-mile long piece expanding by one foot. It will form a shallow isosceles triangle, with base exactly two miles long, and sides 2 miles + 1 foot. How high will the track be at the middle? Less than 1 inch? More than one inch but less than one foot? More than one foot but less than 10 feet? More than 10 feet but less than 20 ft? More than 20 ft?

(2) Tennis balls are frequently sold in packages of 3, with the 3 balls packed into a cylinder. Which is greater, the height of the can or the circumference of the can? By how much? The answer to this question will be much more surprising if the teacher displays such a can. Most students will guess from looking that the height of the can is considerably greater than the circumference.

(3) This is a very common optical illusion. Draw two lines exactly the same length. At the end of one place arrowheads, and at the end of the other reverse arrowheads, something like this:
<-------------------->     >--------------------<
Which line is shorter? By what percentage? While not strictly a math problem, it does show that we cannot always depend on the evidence of our eyes, and that in this case measurement with a ruler is more reliable than vision.

(4) Display two statues. The statues are similar (in the sense of geometry, same shape but different size). Tell the class that both statues are made of the steel (or gold?) that the smaller statue weights 10 pounds, and the larger statue is exactly twice the height of the other. How much would they guess it would weigh?  You could tie this into the use of a scale model of the Titanic in James Cameron's movie. This is also related to the misuse of 3-d figures in statistical graphs, where typically oil consumption might be indicated by the size of an oil barrel, with the height of the barrel proportional to a country's consumption, greatly exaggerating the difference in consumption between different countries.

(5) A rope has been stretched around the Equator of the Earth (25,000 miles, approximately). How much longer does the rope need to be if it is to be raised two feet off the ground and remain a closed loop?

(6) How many people, picked at random, must be in a room before the probability that two or more people share the same birthday is greater than 1/2?

(7) The Monty Hall problem. You are a contestant in Monty's TV show. You are presented with three doors. There is a prize behind just one of them. You choose one door. Without opening that door, Monty opens one of the other two doors that does not hide the prize. (If your first pick was correct so both unpicked doors have no prize, Monty chooses randomly which of those to open.) He says you may stay with your original pick, or switch to the other unopened door. Should you switch or not? How would your choice affect the probability of your winning?