Just why does e appear in so many guises?
This week we pose two interesting e related puzzles:
1) Obtain a large bowl with N strands of spaghetti; grab two loose ends and tie them together. Repeat, until all the loose ends are paired. You will now have a bowl full of loops of spaghetti. On average, what is the expected number of loops?
2) N people walk into a room; each of their (unique) names has been written on a nametag and placed into a bowl. If each person picks a nametag at random, what is the probability that no one gets the right name?
In both cases, the interesting thing is what happens as N increases without bound.
When we were done taping, I remarked to Kyle that, well, surely that’s the end of e related stuff for a while. But I just remembered one of the best e related puzzles of all. We’ll add it here as a bonus:
3) Someone has written counting numbers, one on each of N cards. You don’t have any idea what the largest number is. The cards are shuffled and arranged in a line face down.
You turn the cards over, discarding the cards one by one; you may stop at any time. Your goal is to pick the card with the largest number. (You can’t go back and retrieve a discarded card, and you can’t continue once you stop).
Your strategy, then, is to flip over some number M of cards just to see what the field is like, then taking the first card better than any of the cards in your test sample.
You don’t want M to be too small– you need to get a feel for how big the numbers might be; but you don’t want M to be too big— you don’t want to actually waste the biggest number in your test.
Amazingly, the optimal M works 1/e (almost 37%!) of the time. What is this M, and why does this work?