Can You Solve My Problems?
Alex Bellos wrote a puzzle book Can You Solve My Problems? Ingenious, Perplexing, and Totally Satisfying Math and Logic Puzzles The book contains a mixture of famous puzzles and their solutions. Some of the puzzles are not mathematical in the strictest sense, but still have an appeal for mathematicians. For example, which integer comes up first when you alphabetize all the integers up to a quadrillion?
Recognize the puzzle on that book cover? You’re right! That’s my Odd One Out puzzle. Doesn’t it look great in lights on that billboard in London?
Mine isn’t the only terrific puzzle in the book. In fact, one of the puzzles got my special attention as it is related to our current PRIMES polymath project. Here it is:
A Sticky Problem. Dick has a stick. He saws it in two. If the cut is made [uniformly] at random anywhere along the stick, what is the length, on average, of the smaller part?
The answer to the sticky problem should be the value of the integral from 0 to 1, of the minimum of x and 1-x.30 March 2017, 4:44 am
(The cut is made at point x, and produces a piece of length x and a rest piece of length 1-x.)
This value is 1/4.
The smallest piece will be on the interval (0,50]. Each outcome except 50 is equally likely since there are two ways of cutting the stick to get each numbers on (0,50) while there is only 1 way to cut the stick so that the smaller stick is 50. So, roughly speaking, it seems like it should be 1/4.31 March 2017, 7:45 pm