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Vanishing Socks (Posted on 2024-12-03) Difficulty: 3 of 5
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?

No Solution Yet Submitted by K Sengupta    
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Solution Analytic Solution Comment 2 of 2 |
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.

  Posted by Brian Smith on 2024-12-04 01:58:57
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