How many equal-sized spheres can touch a sphere of the same size? 12, as shown here. Newton guessed this was the maximum, but it was only proved in 1953. Why? There's a bit extra 'wiggle room', so it's a tricky question!
It turns out there's not enough room to stick in a 13th sphere. But there's enough room to take the 5 spheres surrounding any outer sphere and twist them around, one fifth of a turn!
And here's where things get fun. If we keep doing twists of this sort, we get lots of permutations of the 12 outer spheres. How many permutations? Not all of them... just half of them, the so-called 'even' ones. There are
12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 = 239500800
of these.
But now suppose we only do moves like this: first a clockwise twist of this sort around some sphere, then a counterclockwise twist of this sort around some other sphere. Say we keep doing moves like this. How many permutations can we get this way? We get
12 × 11 × 10 × 9 × 8 = 95040
of them. They form a group called the Mathieu group M12, which has lots of wonderful properties.
Now take a deck of cards. Throw out all the black cards and all the aces. Split the remaining deck evenly, so half are in each hand. Shuffle it perfectly, so the cards alternate. Keep doing this as often as you want. You get lots of permutations of the cards this way. How many? This many:
2^11 × 12 × 11 × 10 × 9 × 8 = 2048 × 95040
And here's the really cool part: they form a group that contains M12. Technically, it's the semidirect product of M12 and (Z/2)^11. I just learned this today, from here:
• Persi Diaconis, Ronald Graham and William M. Kantor, The mathematics of perfect shuffles, Advances in Applied Mathematics 4 (1983), 175–196, http://www-stat.stanford.edu/~cgates/PERSI/papers/83_05_shuffles.pdf
Graham is the guy who invented 'Graham's number' - a really huge number I've talked about before. Diaconis started out as a stage magician before becoming a mathematician. He's good at card tricks and has also written lots of math papers about card shuffling.
For Newton and the 'kissing spheres problem', see:
http://plus.maths.org/content/newton-and-kissing-problem
This is where I got the picture here!
For more on Mathieu groups, see:
http://en.wikipedia.org/wiki/Mathieu_group
Puzzle: which sequences of perfect shuffles do you need to get precisely the Mathieu group, and no more?
The answer is hiding in that paper by Diaconis, Graham and Kantor.
It turns out there's not enough room to stick in a 13th sphere. But there's enough room to take the 5 spheres surrounding any outer sphere and twist them around, one fifth of a turn!
And here's where things get fun. If we keep doing twists of this sort, we get lots of permutations of the 12 outer spheres. How many permutations? Not all of them... just half of them, the so-called 'even' ones. There are
12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 = 239500800
of these.
But now suppose we only do moves like this: first a clockwise twist of this sort around some sphere, then a counterclockwise twist of this sort around some other sphere. Say we keep doing moves like this. How many permutations can we get this way? We get
12 × 11 × 10 × 9 × 8 = 95040
of them. They form a group called the Mathieu group M12, which has lots of wonderful properties.
Now take a deck of cards. Throw out all the black cards and all the aces. Split the remaining deck evenly, so half are in each hand. Shuffle it perfectly, so the cards alternate. Keep doing this as often as you want. You get lots of permutations of the cards this way. How many? This many:
2^11 × 12 × 11 × 10 × 9 × 8 = 2048 × 95040
And here's the really cool part: they form a group that contains M12. Technically, it's the semidirect product of M12 and (Z/2)^11. I just learned this today, from here:
• Persi Diaconis, Ronald Graham and William M. Kantor, The mathematics of perfect shuffles, Advances in Applied Mathematics 4 (1983), 175–196, http://www-stat.stanford.edu/~cgates/PERSI/papers/83_05_shuffles.pdf
Graham is the guy who invented 'Graham's number' - a really huge number I've talked about before. Diaconis started out as a stage magician before becoming a mathematician. He's good at card tricks and has also written lots of math papers about card shuffling.
For Newton and the 'kissing spheres problem', see:
http://plus.maths.org/content/newton-and-kissing-problem
This is where I got the picture here!
For more on Mathieu groups, see:
http://en.wikipedia.org/wiki/Mathieu_group
Puzzle: which sequences of perfect shuffles do you need to get precisely the Mathieu group, and no more?
The answer is hiding in that paper by Diaconis, Graham and Kantor.
