The Logical Puzzle - 28 May 2008

Suppose you lay a string on the ground all around the earth right over the equator. The length of the string would be equal to the earth’s equatorial circumference—40,075.02 kilometers. Then, suppose you add 1 meter to the string, and suspend the string directly above the equator, with an even distance from the ground all the way around. Would a cat be able to pass from one hemisphere to another below the string?

Although this puzzle is quite simple, I like it because it’s so counterintuitive. It probably seems inconceivable that adding only 1 meter to such a large circumference would make any noticeable difference in the radius, let alone allow a cat to pass below the string in the space that was added. But if you do the math, you realize that the existing length of the circumference has no significance in determining how the radius would be affected when extending the length of the circumference. Instead, only the length of the addition is significant. The circumference can be expressed as P = 2πr (2 times Pi times the radius). Hence the original radius can be expressed as r = P/(2π). Adding 1 meter to the existing circumference would change the equation to: P + 1m = 2πr, where r represents the new radius. Isolating r, you get: r = (P + 1m)/(2π). Expanding the parentheses, you get: r = P/(2π) + 1m/(2π). Since the original length of the radius was P/(2π), the new radius is 1m/(2π) longer, which is about 16 centimeters (a bit more than 6 inches). That’s enough for a cat to go under and move from one hemisphere to the other.

June’s Puzzle: Josephus Problem

The Josephus problem is an ancient puzzle that involves a group of 41 men standing in a circle. Going around the circle, every second standing man is executed (one skipped, one executed) until only one man is left standing. Assuming that the positions are numbered 1 through 41, which position should Josephus (one of the men) choose if he could, so that he would be the only one to remain standing? Can you generalize the solution for n men? Write a T-SQL solution that returns the position based on the input number of men @n.