It was only a few days until
Christmas, and Santa was dreading
all the stockings he would have to
fill. That night, he had a strange
dream. As he sat staring at a huge
pile of socks of seven different
colors (azure, beige, cabernet,
daffodil, ecru, fuschia, and gold),
the Ghost of Christmas Past
appeared and said, "If you were
to pick two socks at random, the
odds are 50:50 that you would get
a matched pair." He then waved
his hand and all the gold socks
vanished, but the Ghost stated
that the odds of getting a matched
pair were still 50:50.
He waved his hand again, and the fuschia socks vanished, but again he stated that
the odds were still 50:50.
In turn, he made the ecru, daffodil, and
cabernet socks vanish, but in each
case he said the odds of a matched
pair remained at 50:50.
At this
point, Santa counted the remaining
socks and found that he had 25
left. He asked the Ghost how many
socks he had made vanish. The
Ghost replied, "All I'll tell you is
that it is a multiple of the original
number of socks of your favorite
color."
What is Santa’s favorite
color and how many socks did the
Ghost make vanish?
Using a computer is seriously massive overkill. A little bit of algebra will yield a nice recursion.
Let there be D socks in the drawer after the ghost vanishes away S socks of a color; so then the drawer started with S+D socks.
There are S*D ways to make a mismatched pair containing a sock from S, and there are S*(S-1)/2 ways to make a matching pair of socks from S. The union of these two sets of parings represents all the possible parings that disappear when the ghost vanishes S socks.
If we have a 50% probability before and after then we must also have the sets that were removed also have a 50% chance. Then S*D = S*(S-1)/2. This simplifies down to S=2D+1. Then the total number of socks in the drawer beforehand is 3D+1.
From here it is very easy to see a recursion for the total size of the sock drawer D(n+1) = 3*D(n) + 1. We are given D(2)=25.
Then D(3)=76, D(4)=229, D(5)=688, D(6)=2065, and D(7)=6196.
Then the individual quantities of socks are cabernet=76-25=51, daffodil=229-76=153, ecru=688-229=459, fuchsia=2065-688=1377, and gold=6196-2065=4131. For completeness azure and beige are 10 and 15.
The total number of socks vanished is just D(7)-D(2)=6196-25=6171. 6171=3*11^2*17; the only factor appearing among the sock counts is 51=3*17. Then cabernet is Santa's favorite color.