The 50 cards riddle: will this process ever stop? Assume a deck of $50$ cards, numbered from $1$ to $50$. Look at the number of the top card, assume it is $K$. Now, reverse the order of the first $K$ cards of the deck. For example, if $K=5$ change the order of the top $5$ cards so that the fifth card becomes first, the fourth becomes second, etc. Will this process ever stop?
 A: HINT: The process stops only when $1$ comes to the top of the deck. Suppose that you knew that the process would eventually stop with a deck of only $49$ cards. Now start with your shuffled deck of $50$ cards. 


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*If card $50$ is on the bottom, you’re essentially playing with just the top $49$ cards, and by hypothesis the $1$ will eventually come to the top.  

*If card $50$ is not at the bottom, but card $1$ is, pretend temporarily that card $50$ is really card $1$. Until it comes to the top you’re playing with the top $49$ cards, and since card $50$ is filling in for card $1$, by hypothesis it will eventually come to the top. At that point you’ll reverse the whole deck, bringing $1$ to the top.  

*If neither card $50$ nor card $1$ is on the bottom, you’re still essentially playing with the top $49$ cards until either the $1$ comes to the top, in which case you can stop, or the $50$ comes to the top. In the latter case you’ll reverse the deck, putting the $50$ on the bottom, and you’re in the first case.


So how could you show that the process always eventually brings the $1$ card to the top when there are only $49$ cards?
