(Since we’ve been offline for a week or so, due to a tremendous ice storm that has paralyzed the town, we add a special bonus: the very first Math Factor episode ever aired, from January 25, 2004.)

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?

Amazingly, an optimal path for the Princess is to swim in a half circle of radius 1/8 that of the lake, then dash out to the edge.
We’ll give an analytic proof, but we could give a totally synthetic (geometric) proof as well.

An Analytic, Calculussey Proof

First of all, we should ask:

At a given moment, how fast can the Princess swim towards the edge of the lake?

Let’s not worry about our actors changing directions— this doesn’t really affect our thinking (remember we assumed they could change directions instantly). So the Beast will be dashing around the lake and the princess taking some route to get away.

Let’s scale things so that the lake has radius 1, and the Beast’s speed is 1. At time t then, he will have travelled a distance of t, and the Princess will have traveled one-fourth as far, ¼t.

At a given moment, how fast can the Princess swim towards the edge of the lake?

Consider what happens over a very very small interval of time, of size Δt, when the Princess is r away from the center of the lake.

The Beast travels Δt along the shore. The Princess has to keep the beast on the opposite side of the lake, and so has to swim proportionally less,
r Δt around.

How much can she increase r, if she is travelling a total of ¼ Δt, and needs only to go around a distance of r Δt? It is well worth learning how to think like an 18th Century Mathematician: essentially we have a right triangle, and can use the Pythagorean Theorem. (We don’t really have exactly a right triangle, but the difference is negligible, and as Δt decreases, is less and less important, until irrelevant in the limit.)

By the Pythagorean Theorem, an increment of change of radius Δ r is √( (¼ Δ t)² – (r Δ t)²)

(In the limit, this is good enough; all the differences are vanishingly small compared to Δ, as Δ decreases.)

In other words, in the limit,

dr = √ (¼²-r²) dt

Using a little calculus, and remembering that at t = 0, we have r = 0 as well, we can check that

r = ¼ sin 4t

Remember that t is not only measuring time, but also the monster’s distance around the lake; the lake is radius 1, so this is also the radian measure around the lake. In other words, the initial part of the Princess’ path, in polar coordinates is:

r(θ) = ¼ sin 4θ

a circle as promised!

Amazingly, there are as many rational numbers as counting numbers!

We also discuss Zeno’s paradoxes and a famous summation problem.

We answer last week’s puzzle about running animals and ask about the average speed of our old junker.