The Two Piles

The first "fix" the magician has set up is that the deck will be stacked with red and black cards alternating from top to bottom. The second "fix" the magician has arranged is that the deck is split so that each stack has a different bottom color card. The illustration above portrays the deck without numbers or suits and so that one can see the colors of the cards without the backs obscuring them. Note that this splitting of the deck puts an odd number in each stack. The gray edged cards are one stack the white edged cards the other stack.

The Riffle Shuffle

The riffle shuffle does not change the order of color of either stack. It merely combines them so that some of one stack are followed by some of another stack. As we go through the deck the victim will be picking the first card and all odd number positions in the deck. The magician's first card will be the second card and he will have all the even number positions in the deck.

The Inserted Cards Beginning

We will assume that the gray edged stack has white edged cards inserted into it. Reading from left to right we will start by inserting some of the white edged cards. Obviously, the first card to be inserted is going to be red. If it goes immediately after a black card the order of alternating red and black won't be affected. The cards will still alternate black and red. If an even number of white edged cards are inserted the alternation of red and black won't even be detected. Note: The striped cards are the magicians cards. Each of his red cards will naturally follow a black card. After he knows the victim's card is black he merely predicts red.

However if an odd number of cards is inserted order will be changed but the first card out of order will be the victim's. All of a sudden he gets a red card. Again the striped cards are the one the magician gets. When he sees the victim get a red card he switches and predicts black. He won't be wrong.

If the first red white edged card is inserted after a red card things start with a bang. Unfortunately, for the victim, the surprise red card is one he gets. This is because in the gray stack the red cards are even so to insert a card after a red card means the inserted red card will be the victim's. This warns the magician that he needs to predict black for the next card. If an odd number of cards are inserted the following cards will continue to alternate without interruption.

However, if an even number of cards is inserted the alternation of red and black will be interrupted. But again this will only affect the victim.

Rigorous Proof?

One begins to see the pattern here. After a riffle shuffle of two deck with alternating colors that start with different colors there will never be more than two cards in a row of the same color and "surprises" in color will occur in odd numbered positions.

If you want more proof you'll have to work on this some yourself. After you have suffered a while, get in touch with me and we'll try and get to the bitter end together. Send your proofs or questions to Juan Wilson. Have fun!