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In a standard deck, there are 13 cards in each suit,such as hearts. For simplicity we'll number them from 1 to 13. This question is about these 13 cards.

Start with the cards in the usual order. This time you are allowed to divide the deck into three piles, consisting of the top n cards, the next m cards, and the remaining 13-n-m cards. Keeping the order within these piles unchanged, you place one pile at the top, one in the middle, and one on the bottom. You may repeat this as many times as you like, changing m and n if you wish. How many arrangements can you get?

My answer:

So we can number them from 1 to 13 in the usual order: 1 2 3 4 5 6 7 8 9 10 11 12 13

So as one case I can have firs pile=4 cards, second pile=4 cards and last pile=5 cards. If I do the described permutation, I get:

1 5 9 2 6 10 3 7 11 4 8 12 13

Is this correct? If this is correct, then this question is asking me to consider different values of m and n and consider different arrangements based on these values?

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    Does "this time" in your question indicate that you had to do something similar "last time"?2011-03-08
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    @Fabian: I am not sure what you mean.2011-03-08
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    By the way: why you get `1 5 9 2 6 10 3 7 11 4 8 12 13`? Maybe I misunderstand the question, but when I take $n=m=4$ and I put the last pile on to, the second pile in the middle, and the first pile at the bottom, I get `9 10 11 12 13 5 6 7 8 1 2 3 4`.2011-03-08
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    I thought that if I take n=m=4, I have first pile=4 cards, second pile=4 cards and last pile=5 cards. So I place first card from first pile at the top, then first card from second pile below that, then first card from third pile below that, and so on. But now I am not sure that I understand the question.2011-03-08
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    I read the question that you have to shuffle the whole piles but not the individual elements.2011-03-08
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    You are right but I thought that it does not make sense. I explain why. First pile which is the top one contains: 1 2 3 4, the middle pile contains 5 6 7 8 and bottom pile contains 9 10 11 12 13, so the question says that "Keeping the order within these piles unchanged, you place one pile at the top, one in the middle, and one on the bottom", so If I do them in order I get identity (just what I started with), I meant if I place the first pile at the top, second pile at the middle and third pile at the bottom. Can you please enlighten me about your understanding of this question?2011-03-08
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    you can shuffle the three piles but not the cards inside the piles...2011-03-08
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    So are you allowed to shuffle the piles in any order? but when the question said "Keeping the order within these piles unchanged, you place one pile at the top, one in the middle, and one on the bottom.". I thought that it means shuffling piles in order so you place top pile at top, middle pile at the middle and bottom pile at the bottom.2011-03-08

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