Comments on: 8 perfect shuffles https://blog.yhuang.org/?p=84 here. Tue, 14 Oct 2025 11:10:14 +0000 hourly 1 http://wordpress.org/?v=3.1.1 By: me https://blog.yhuang.org/?p=84&cpage=1#comment-96199 me Wed, 23 Mar 2011 21:27:31 +0000 http://scripts.mit.edu/~zong/wpress/?p=84#comment-96199 If these 16 shuffles are in-shuffles, where the first card goes on top each time, then the top card is the same all the time, and there would be no problem to solve. So let's assume you're talking about out-shuffles, where the first card in the second half of the deck goes on top. For this, you apply a modified version of the first result from the post. Card n in the deck is sent to 2^s*(n+d/2) (mod d-1) in this case. So you are solving 2^16*(n+27) = 0 (mod 53) for n in {0,...,53}, the solution of which is n = 26. This corresponds to the first card of the third suit, so in your case the ace of hearts. If these 16 shuffles are in-shuffles, where the first card goes on top each time, then the top card is the same all the time, and there would be no problem to solve.

So let’s assume you’re talking about out-shuffles, where the first card in the second half of the deck goes on top. For this, you apply a modified version of the first result from the post. Card n in the deck is sent to 2^s*(n+d/2) (mod d-1) in this case. So you are solving 2^16*(n+27) = 0 (mod 53) for n in {0,…,53}, the solution of which is n = 26.

This corresponds to the first card of the third suit, so in your case the ace of hearts.

]]> By: Rits https://blog.yhuang.org/?p=84&cpage=1#comment-93154 Rits Thu, 30 Dec 2010 21:10:23 +0000 http://scripts.mit.edu/~zong/wpress/?p=84#comment-93154 Hey, Can you help me with this problem? A new deck of cards is in the standard order from the top: ace of spades, deuce of spades, three of spades…until king of spades. After that all the diamonds follow in the same order, then all the hearts, then clubs. Finally, there are two jokers on the bottom. After performing 16 perfect shuffles, what number is on the top card now? Hey,
Can you help me with this problem?

A new deck of cards is in the standard order from the top: ace of spades,
deuce of spades, three of spades…until king of spades. After that all the diamonds follow in the same order, then all the hearts, then clubs.
Finally, there are two jokers on the bottom. After performing 16 perfect shuffles, what number is on the top card now?

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