🚀 go-pugleaf

I need help on a sphere packing question.  Suppose you have a white ping
pong ball, and 13 red ones, all the same size.  Can you glue all of the red
ping pong balls to the white one?  In other words, is it possible to place
13 unit spheres around one unit sphere, so that each of the 13 is touching
the central sphere, and none of the 13 intersect each other?

It is a simple matter to do it with 12.  Simply place the spheres at the
corners of an icosahedron centered at the center of the white ping pong ball.
It turns out this leaves some extra room for spheres to be moved around, but
is it possible to move them around so that another sphere can be squashed
in there?

In the meantime, I leave you with two simpler questions:

1.  Suppose you pick a number at random from 0 to 1.  Assume uniform
    distribution (ie, pdf = 1 from 0 to 1).  Clearly, that number is
    not going to be greater than 1.  Now you pick another number at
    random from 0 to 1.  The probability that the sum of these two
    numbers is greater than 1 is simple to calculate; it is 1/2.  If
    it is greater than 1, stop; otherwise, continue picking numbers
    at random until the accumulated sum is greater than 1.  Derive the
    average number of picks required.  (What is the answer if the
    distribution is arbitrary?)

2.  Suppose I have a deck of cards numbered from 1 to 100.  You and I
    are dealt two hands of 50 cards each, randomly, from the deck.  We
    both sort the cards in increasing order.  In the first round, we
    both play our first card; the player with the higher card wins
    one dollar, and the two cards are discarded.  This routine is
    repeated for our second card, third card, etc.  What is the
    probability that I win this warped version of War by a score of
    50 to nothing?

byron elbows
(mail to brian@cs.ucla.edu)