Gilbreath Principle

The Gilbreath Principle Jonathan Bredin September 24, 2001 1. Gilbreath principle (a) Arrange Arrange cards so that colors colors alternate alternate (b) Cut the deck so that the bottom of each half is a different different color color (c) Riffle shuffle shuffle (d) Remove Remove the cards in pairs from the top. Each Each is of different different color. (e) Why? Why? i. ii. iii. iv.

assume assume that first card to hit table is red if next card comes from same side, it’s it’s black other other side is also black in any case, the bottoms bottoms of the halves halves are still different different colors

2. more general general Gilbreath Gilbreath (a) order deck in suits (spades, (spades, hearts, clubs, diamonds, diamonds, spades, hearts, hearts, etc). etc). (b) deal cards to form a pile (reversin (reversing g order) (c) riffle shuffle shuffle (d) draw quadruple quadruplets ts - will get sets of all four cards cards (e) can extend to two entire entire decks decks 3. non-messing-up theorem (a) shuffle shuffle (b) deal cards cards face up to form a rectangle rectangle (c) sort each row in increasing increasing order (d) sort each column column in increasin increasing g order (e) notice notice that rows are still ordered ordered (f) why? 4. dilution dilution (a) dived dived deck into red and black black halves (b) take take  n  cards from black half and put into red (c) shuffle shuffle red half  (d) take take  n  cards from red half and put into black (e) does the red half have more red cards than the black half has black black cards? 5. Monty Monty Hall Hall (a) guess which which of three cards cards has the ace (b) reveal reveal which non-guesse non-guessed d card does not have have an ace (c) what is the best strategy strategy to find the ace now? 1

(d) switch gives p=2/3 6. Derivative (a) deal three cards (b) two cards of same color (c) what is probability that the third card is that same color? (d) 1/4 (e) sucker bet - give even odds 7. Parity (a) take three red cards from the deck (b) put one red one back, take three black cards (c) put one black one back, take three red (d) repeat (e) can there be the same number of red and black cards? (f) no, parity (g) always holding an odd number of cards 8. Pick four (a) place cards face down ace-9 in increasing order (ace left) (b) remove a card from either end (c) remove a card from either end (d) remove a card from either end (e) add value of three cards and divide by 6 =  n (f) turn over the nth card left-right (g) it’s the four 9. find unknown card (a) divide 21 cards into 3 equal groups (b) observer chooses 1 card from a group, the card is secret but the pile is not (c) place the chosen pile between the other two (d) deal out 3 piles of seven again (e) ask which pile the chosen card is in (f) place chosen pile in between others (g) deal again (h) identify pile (i) place chosen pile between other two (j) secret card is the 11th at the top of the deck (k) why? i. if we pick a random card in a pile, say it is the  ath in the order of the pile ii. then we place it in the 7+ a-th slot when we stack the other two piles iii. then the card is in the   (7 + a)/3 + 1 position in the new pile.... iv. at the end the card is in the position

 7+3  + 1 + 7  + 1 3 a



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v. the possible values for the inner floor are 2, 3, or 4 vi. this means that the possible values for the next floor value are 3 (the whole part of 10, 11, 12 divided by 3) vii. add one so our chosen card is fourth in its stack viii. there are 7 cards ahead of our card once we stack the piles ix. so the chosen card is the 11th

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