Another pursuit puzzle:

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Three crazed, robotic professors (or, if you prefer, “spiders”) try to chase down a psychic, but slightly faster student (the “fly”) along the edges of a tetrahedron. It’s easier, perhaps, to draw it out in the view at right below.

Is there a strategy that allows the professors to catch their prey?

Wait a sec– did we say the prey is SLOWER? Obviously one fast hunter can chase down slower prey, even if the prey knows exactly what’s coming…

Wait another sec– isn’t the point that the professors are morons and robotic? Maybe we DID state the puzzle correctly.

Long time listeners know it wouldn’t be the first time we’d inadvertently added a “meta-puzzle”, namely, to figure out what it is we’d meant to say!! Thanks Byon for pointing that out!

(PS Don’t forget the spoiler tag if you post spoilers in the comments!)

(PPS Apologies: our policy of holding back solutions sometimes gets the order of the comments pretty scrambled)

We’ll have some pursuit puzzles over the next couple of weeks; this segment’s puzzle has a simple and elegant solution, but it might take a while to work it out!

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In the meanwhile, here’s a little discussion about the glass of water problem.

Each time we add or subtract 50%, we are multiplying the quantity of water by 1/2 or 3/2. If we began with 1 glass’ worth, at each stage, we’ll have a quantity of the form 3^m/2ⁿ with m,n>0  Of course that can never equal 1, but we can get very close if m/n is very close to log₃ 2 = 0.63092975357145743710…

Unfortunately, there’s a serious problem: m/n has to hit the mark pretty closely in order for 3^m/2ⁿ to get really close to 1, and to get within “one molecule”s worth, m and n have to be huge indeed. 

How huge? Well, let’s see: an 8 oz. glass of water contains about 10²⁵ molecules; to get within 1/10²⁵ of 1, we need m=31150961018190238869556, n=49373105075258054570781 !!  One immediate problem is that if you make a switch about 100,000 times a second, this takes about  as long as the universe is old!

But there’s a more serious issue.

In a glass of water, there’s a real, specific number of molecules. Each time we add or subtract 50%, we are knocking out a factor of 2 from this number. Once we’re out of factors of 2, we can’t truly play the game any more, because we’d have to be taking fractions of water molecules. (For example, if we begin with, say, 100 molecules, after just two steps we’d be out of 2’s since 100=2*2*some other stuff.

But even though there are a huge number of water molecules in a glass of water, even if we arrange it so that there are as many 2’s as possible in that number, there just can’t be that many: 2⁸³ is about as good as we can do (of course, we won’t have precisely 8 ounces any more, but still.)

If we are only allowed 83 or so steps, the best we can do is only m= 53, n = 84 (Let’s just make the glass twice as big to accommodate that), and, as Byon noted, 3^53/2^84 is about 1.0021– not that close, really!

So how close, and how quickly, can we get back to exactly one glass of water, adding and subtracting 50% of the total at each step. And what is happening with the “reverse of the square is the square of the reverse” property of 2012, 2011 and 2010?

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In which we continue our contest for SOME interesting fact about the number 2012, describe Newton’s Law of Cooling, and ask another puzzle on the mixing liquids.

We HAVEN’T yet fully answered the coffee and cream question: a follow up post will be coming soon!

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In which we confess further delight in arithmetic…

1) Send us your candidates for an interesting fact about the number 2012; the winner will receive a handsome Math Prize! As mentioned on the podcast, already its larger prime factor, 503, has a neat connection to the primes 2,3,5, and 7.

2) So what is it about the tetrahedral numbers, and choosing things? In particular, why is the Nth tetrahedral number (aka the total number of gifts on the Nth day of Christmas) is exactly the same as the number of ways of choosing 3 objects out of (N+2)? Not hard, really, to prove, but can you find a simple or intuitive explanation?

3) Finally, about those M&M’s. Maybe I exaggerated a little bit when I claimed this problem holds all the secrets of the thermodynamics of the universe, but I don’t see how! Many classic equations, such as Newton’s Law of Cooling or the Heat Equation, the laws of thermodynamics, and fancier things as well, can all be illustrated by shuffling red and blue M&M’s around. What I don’t understand is how anything got done before M&M’s were invented!

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How much does my True Love love me truly?

Kyle and Chaim ponder the question…

[spoiler]

This is, of course, the on air version of last year’s post, The Nth Day of Christmas, when the podcast was quiet. But don’t peek!

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Quick answer soon, and in the New Year, a longer discussion on the merits of mixing cream with your coffee. In the meantime,

Peace on Earth, for all living things,

Chaim and Kyle

Harry Kaplan joins us for discussion of cake and coffee– and leaves us with a counter-intuitive puzzle…

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A quick hello from Chaim and Kyle as the Math Factor returns!

We’ll be the first to say our Coffee Pot Question isn’t our deepest puzzle ever, but it sure did make a difference in Chaim’s life!

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When I became chairman of my department a few years ago, I moved from my office far down at the end of the hall to one much closer to the center of action: the tea room! I make a lot more visits there than I used to, and began to notice a frustrating pattern:

Far more often than seems reasonable, there’s not even a full cup of coffee in the coffee pot! Once again, someone has left a nearly empty pot with no regard to the next person (me, whine)!

This seems to happen so often I began to wonder what kind of boors I’ve been working with all these years. They seem like nice people and all, but…?

And then I realized: there’s a perfectly logical reason, a mathfactor puzzle, if you will, that explains this phenomenon perfectly, no boors required, no special tricks, just sensible activity by all. My faith in my colleagues has been restored.

Why is it that on average I see an emptier rather than fuller coffee pot?

PS let us know what works for when we return…

Well, it’s been quite a while, and I guess the important news is that the M.F. is coming back soon! Look for new segments starting around the end of the year!

All those encouraging emails, prodding emails, whining emails have had their effect—but I have to say, in the end, it was the Juniors Fabulous cheesecake that probably made the difference.

Meanwhile, a steady accumulation of great ideas and on top of that, when we do come back, we’ll be listed on the National Science Foundation’s app/site, Science 360. Thanks to all those that have hung around waiting—your patience is valued. 

One question: Over the years we’ve experimented with a lot of formats (short, long, interviews, deep stuff, light stuff, book reviews, puzzles, Kyle and me shooting the breeze.) What format works best for you?