## EB. Busy Beavers and Dumb Robots

Those dumb robots can do anything! Anything at all, that any computer can do.

Those dumb robots can do anything! Anything at all, that any computer can do.

One of the great discoveries of the twentieth century is that mathematics can describe the limits of mathematical thought! We’ll discuss some of these ideas from time to time in coming weeks. In this segment, we consider Alan Turing’s insightful question:

*Can the answer to any mathematical question be computed?*

We explore Barry Cipra’s Tag Deal a bit more…

We catch up with Raymond Smullyan, author of many fantastic books on logic, puzzles and paradoxes at this year’s Gathering for Gardner!

We discuss, among other things, whether all mathematicians are liars.

Send us your favorite paradoxes of this kind and we’ll report back on April 15.

We consider that perennial spring conundrum: Would a woodchuck chuck her own wood if she would chuck wood for exactly those woodchucks who would not chuck their own wood?

What follows after 0, 1, 2, … , once you’ve managed to list every counting number?

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*.

As B Boom wrote, the first pirate can make a proposal that gives him all but 49 (about, depending on the rules) pieces of

the gold. Read the rest of this entry »

Amusingly, this problem has exactly the same solution as the proof that there are as many rational numbers as there are counting numbers. And the proof generalizes: one stork can catch three frogs, or ten or fifty.

Here are some bonus problems:

- The stork can catch the frog even if it can start at any
*rational*number and hop any fixed*rational*distance each step. - 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.

A contestant for our Million-Dollar-Give-Away sent in **Rayo’s Number**, hitherto the largest number ever used for any real purpose: to wit, winning the

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.