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?

1 + 1/2 + 1/3 + 1/4 + . . .
diverges to infinity, growing as large as you please!

If you try adding terms up on a calculator, this scarcely seems possible! By the time you have added the hundredth term, you will have a reached only a whopping 5.187… (and each new term will be less than .01).

After adding up a million terms, you will have made it only to about 14.39272672… — and each new term will be less than .000001. Does the series really diverge?

The eighteenth century mathematician Jacob Bernoulli gave a very elegant proof that it does:

1/2 is at least 1/2

1/3 + 1/4 is at least 1/4 + 1/4 = 1/2

1/5 + 1/6 + 1/7 + 1/8 is at least 1/8 + 1/8 + 1/8 + 1/8 = 1/2

1/9 + . . . + 1/16 is at least 8 x 1/16 = 1/2

etc. So the result of adding up the first 2ⁿ terms 1/2 + 1/3 + . . . + 1/2ⁿ is at least n/2, and in particular, can be as large as we please.

But this does take a long time to get anywhere. To add up to, say, 100, Bernoulli’s proof shows us that 2¹⁹⁸ (about 4×10⁵⁹) terms will suffice. But maybe this is more than we actually need.

A basic fact from calculus is that the area under the curve y = 1/x, from x = 1 to x = N is exactly ln N.

Now the area of a box 1 unit wide and 1/n units tall is 1/n, and boxes of width 1 and heights 1, 1/2, 1/3, . . . altogether have area 1 + 1/2 + 1/3 . . .

Here we see that these boxes can be arranged to show that

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 > ln 8

Shifting the boxes over, we see that

1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 < ln 8

This gives us a much better bound on the harmonic series. Generally, we have that

1 + 1/2 + . . . + 1/n is between ln (n+1) and (ln n) + 1.

So to be sure that the series sums to at least 100, we can be sure that e^100-1 (about 2.7×10⁴³) terms will suffice!

So the sum of the first million terms is about (ln 1,000,000) + γ, and if we want to sum to 100, we need to have n such that ln n + γ is greater than 100; in other words, e ^{(100 – γ)} (about 1.5×10⁴³) terms will do.

The series Σ 1/(n ln n) diverges even more slowly still, taking about e^e^n terms to sum to n (!!) The series Σ 1/(n (ln n) (ln ln n)) takes e^e^e^n terms to sum to n. Etc!!

Dana Richards, editor of Martin Gardner’s Colossal Book of Short Puzzles and Problems explains why the worm makes it, in only about 15,092,688,622,113,788,323,693,563,264,538,101,449,859,497 steps! (Give or take a few.) This incredible fact depends on the mysterious Harmonic Series, discussed a little more in our next post.

Dana Richards, editor of The Colossal Book of Short Puzzles and Problems discusses the amazing Martin Gardner and his legacy!

Around 1875, Georg Cantor created — or discovered if you like — the transfinite ordinals : the list continues 0, 1, 2, …, then ω , ω + 1, ω + 2, etc, for quite a long long way. John H. Conway tells us about his Surreal Numbers , which add in such gems as

1 / √ ω

Check out Knuth’s Surreal Numbers, Conway & Guy’s Book of Numbers , or for more advanced users, Conway’s On Numbers and Games.

And we explain how the Stork can catch the Frog.

Here are some bonus problems:

1. The stork can catch the frog even if it can start at any rational number and hop any fixed rational distance each step.
2. However, if the frog can start at any real number or hop any real distance, the stork has no strategy that guarantees a catch. This is, in effect, the same as proving that the real numbers are not countable.
   

LARGE NUMBER CHAMPIONSHIP

Check out the article by Scot Aaronson that inspired them to duke it out! And this thread on the math forum is quite interesting as well.