HG. Two Love
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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Louis Bookbinder said,
January 4, 2012 at 12:07 am
I couldn’t figure the connection between 503 and the first four primes, until I accidentally cubed each of them and found that 503 is the sum of all 4 cubes! About the M&Ms – the first swap of 50 M&Ms makes each pile (bag, box, etc) 90% as pure as it was before (red becomes 90% red and blue becomes 90% blue). At this point I assume the M&Ms are uniformly mixed in each pile. The next time, 81% of the original color is kept but 10% of the original color which was taken away the first time is returned so the pile is 82% its original color. This continues with the colors getting more and more evenly mixed until about 20 swaps when each M&M scoop is, on average, half red and half blue. This is exactly the same as the old puzzle about moving a teaspoon back and forth between a barrel of water and a barrel of wine (each barrel of same capacity) and eventually both barrels are half water and half wine as near as you can measure (within a few micrograms). I can’t come up with an equation to measure how fast this occurs, however.
Aleksandra said,
January 5, 2012 at 2:21 pm
Hi! Love the podcast . I was wondering, maybe you could explain in a simple way Birch and Swinnerton-Dyer conjecture. I’ve been reading about this but cannot understand it. Btw. I’m 16 so maybe this is the reason. = )
I’m very glad you record this podcast.
Aleksandra, Poland