Northwest to the Pole And Two-Headed Coins
Old Northwest challenge (Eric Brahinsky)
Suppose you start at the earth's equator and travel continuously northwest until you reach the North Pole. What does your path look like? How long is your path?
The path is about 9,000 miles long. A tiny right triangle with base 1 and height 1 has hypotenuse (12 + 12) = (2) ("Pythagorean Theorem"). Therefore as you head northwest, you go (2) times as far as your progress northward.
Since the distance due north from the equator to the North Pole is a quarter of the earth's circumference of about 25,000 miles, the distance traveled is 2 x (25,000/4) or about 9,000 miles. Since near the pole the distance around the pole gets very short, at the last minute the path actually spirals around the North Pole an infinite number of times!
Joel Earnest-DeYoung concluded that "the last meter would be very dizzying." He, George Murray, and Clint Staley actually computed a formula for the traveler's increasing western longitude y as a function of the latitude x: y = log tan (x/2 + /4) (in radians).
New two-headed coin challenge (George Murray)
You have a fair coin and a trick coin with two heads. You pick one at random, without noticing which one. The first time you toss it, it comes up heads. What is the probability it will come up heads the second time? (What if you don't actually see the first toss, but three of your friends, each of whom is right two-thirds of the time, all say it was heads?)
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