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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?

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Eric Demaine of MIT will control your minds across all time and all space!

Not surprisingly, we suppose, this trick is closely related to an important mathematical tool of the telecommunications industry:

Gray Codes are tremendously useful; a great, very readable discussion is in Chapter 2 of Martin Gardner’s Knotted Donuts and Other Mathematical Entertainments.

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Now, really, tell me, what good is a podcast if you can’t promote your beautiful new book?

We are very very pleased to announce the publication of The Symmetries of Things, a comprehensive, modern account of the mathematics of symmetry, complete with over 1000 illustrations!

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The Collatz function on the counting numbers is really quite amazing: Divide by 2 if you can, otherwise multiply by 3 and add 1. Iterating this seems always to lead to the loop … 4, 2, 1, 4, 2, 1

For example: 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1 → 4 → 2 → etc.

Does this always happen??

Dunno. No one does. But it is known that you will eventually loop if you start with any number up to about 5 x 10^19

(we accidentally exaggerated this in the podcast).

Try it for 27 for a daunting peek at the difficulty of this problem!

Neil Sloane of ATT Labs shares some his favorite integer sequences from his online encyclopedia!

Recaman’s Sequence is especially perplexing! Sloane asks: does every number eventually appear?

(No one yet knows the answer!)

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

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Mathematics writer Barry Cipra shows us Tag Deal, a simple but perplexing puzzle with cards.

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We present a recording of Raymond Smullyan’s lecture at the Gathering for Gardner, March 30, 2008; Newcomb’s paradox really is a stumper.

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

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