Gilbreath Principle

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The Gilbreath Principle is the fact that when a sequence (which may or may not be a repeating sequence) is riffle shuffled into its reverse, there are certain properties of the mixed outcome that are mathematically guaranteed.

The observation was analyzed and documented by Norman Gilbreath and first described in The Linking Ring, Vol. 38, no. 5, July 1958, page 60 under Magnetic Colors.

Eight years later, in the June 1966 Linking Ring, he published a generalization, extending his first observation from the case n = 2 to arbitrary n, which is referred to as the Second Gilbreath Principle.

Applications Published in Genii

  • STRANDBERG, Per - Matchmaker, published in Genii 2009 August. The magician shows a deck to be well mixed, then places it face down before a spectator, inviting him to cut off the top half or so and table it face up beside the face-down (bottom) half. He openly Riffle Shuffles the face-up block into the face-down block, interlacing the cards, if not perfectly, at least thoroughly, then he deals the top two cards of the deck as a pair, without reversing their order, in front of the spectator. He deals the next two cards as a pair in front of himself. Each pair might be comprised of a face-down card and a face-up card, or two face-down cards, or two face-up cards, depending on how the deck was initially cut and shuffled. He continues dealing pairs in front of the spectator, then himself, alternating until the deck is exhausted, then he says, "Let's take away all the pairs that aren't comprised of one face-up card and one face-down card." Suiting actions to words, he removes all the pairs that consist of either two face-down cards or two face-up cards, placing them in a pile on the side, as he directs the spectator to do the same with the pairs in his own row. Finally, he says, "Now let's play a game—if a pair that's left is two different colors, red and black, you get one point. If it's two cards the same color, you get two points. If it's two cards the same color and matching suits, you get ten points. Let's see how you did." The magician turns over the face-down cards in each of the spectator's pairs, adding up the points as he goes. All the pairs turn out to be matching colors (worth two points each), but there are no matching suits. The magician says, "Not bad—let's look at mine," then turns over all his face-down cards, showing every one of his pairs to match in suit, making them worth ten points each and giving him the clear victory.
  • TARCHINI, Dr. Giorgio - Fresh Gilbreath, published in Genii 2009 March. A new discovery pertaining to the Gilbreath Principle that allows the magician to have a spectator deal eight cards from a thoroughly shuffled deck face down onto the table, whereupon the magician accurately divines whether each card is red or black.
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