I like the problem, but his explanation on counting the number of loops of length 100 seems wrong.
In the first box, you only have 99 choices (you can't choose itself or you have a loop of 1 which is < 100).
The second box has 98 choices (can't choose itself plus the previous box# or you have a loop of 1 or 2 which is < 100).
etc.
So # of loops of length 100 is 99!
# of unique loops of length 100 is 99!/100.
P(L=100) = (99!/100)/(100!) = 1/10000.
I assume I made a mistake somewhere, but I can't see it. Any ideas?
I like the problem, but his explanation on counting the number of loops of length 100 seems wrong.
In the first box, you only have 99 choices (you can't choose itself or you have a loop of 1 which is < 100).
The second box has 98 choices (can't choose itself plus the previous box# or you have a loop of 1 or 2 which is < 100).
etc.
So # of loops of length 100 is 99!
# of unique loops of length 100 is 99!/100.
P(L=100) = (99!/100)/(100!) = 1/10000.
I assume I made a mistake somewhere, but I can't see it. Any ideas?
The first box (the one with your number) is the first choice in the loop, or it will be the last choice because the 99th box will contain your number. Either way it's 100! no 99!
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The first box (the one with your number) is the first choice in the loop, or it will be the last choice because the 99th box will contain your number. Either way it's 100! no 99!
Ok, I had a huge post typed out explaining why I still thought something was wrong...then it hit me.
I was trying to compute the number of ways the numbers inside the boxes can be arranged. This gets complicated, and is not 100!, but does end up being 100!/100 (ie. 99!) right off the bat. That's why I got 99! in my original reply. I shouldn't have divided that number by 100 since the duplicates are already removed.
What I should have been computing is the number of ways the boxes themselves are arranged (ie. the order in which the boxes are opened within the loop) which gives 100!. Removing duplicates gives 100!/100.
Happy July 4th, time for my annual posting of this song.
Fun fact: Real American was wrote and recorded by Rick Derringer, most famously known for the greatest One Hit Wonder song of the 70s, Rock & Roll Hoochie Koo. Rick wrote a *ton* of great songs after, mostly for other artists.
He has since been plagued with Trump Brainworms, but this is not surprising.
Keeping in theme with the thread, here's a great video about it: