You can either select "draw arrow" under the "Mouse click to" drop down menu, or hold down the ctrl key while clicking to draw arrows. For instance, if the first column of your table had a 1 over a 3, you should draw an arrow from the circled 1 to the circled 3. You can tell this from looking at the table. Your goal is to draw an arrow from the circle with the number a to the circle with the number b if the permutation takes a card in position a to position b. This will draw some circles with numbers in them. This will show the table for the chosen type of shuffle. To begin, press "New Random Shuffle" or "Copy Riffle Shuffle Above". Going back to our three card deck where we move the top card to the bottom, we would represent this as:īelow, you can experiment with various presentations for riffle shuffles: In the first deck to its new position in the second deck. In these, we draw two decks of cards side by side and draw a line from a card's original position We can also represent these permutations using function diagrams. Each column will have a card's original position in the deckįor instance, if we had a three card deck and moved the top card to the bottom, we would represent this as:īecause the first (top) card moves into the third (bottom) position, the second card moves into the first (top) position, and the third card moves into the second position. We'll have two ways of presenting a reordering.įirst, we can present a way of reordering the card as a table. To answer this question, we need some way to talk about deterministic ways of reordering decks of cards. Or have we just chosen a particularly bad choice of fixed shuffle? One might wonder if the problem here is caused by the fact that we're shuffling the cards the same way each time, 
Keep playing with the interactive until you're convinced that perfect riffle shuffles aren't a great way of shuffling cards. The first couple of shuffles seem to mix up the deck fairly well, but what happens after many shuffles?
Change the "Color Palette" to "Rainbow 1" to color each card differently to make it easier to follow each card.Press "stop" and move the slider to manually watch the animation.The interactive below will allow you to visualize perfect riffle shuffles: There are two possible ways of interleaving the two halves, one (an in-shuffle) where the top and bottom cards wind up inside the deck,Īnd the other (an out-shuffle) where the top and bottom cards wind up in the same location. Perfect RifflesĪ riffle shuffle is performed by taking a deck of cards, dividing the deck in half ("cutting" the deck), and then interleaving the two halves. In the process, you will hopefully be convinced that riffle shuffling is actually a fairly good technique for shuffling cards. In this series of interactives, we will explore various mathematical topics related to shuffling decks of cards.
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