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ÿþThe Gilbreath Principle Jonathan Bredin September 24, 2001 1. Gilbreath principle (a) Arrange cards so that colors alternate (b) Cut the deck so that the bottom of each half is a different color (c) Riffle shuffle (d) Remove the cards in pairs from the top. Each is of different color. (e) Why? i. assume that first card to hit table is red ii. if next card comes from same side, it s black iii. other side is also black iv. in any case, the bottoms of the halves are still different colors 2. more general Gilbreath (a) order deck in suits (spades, hearts, clubs, diamonds, spades, hearts, etc). (b) deal cards to form a pile (reversing order) (c) riffle shuffle (d) draw quadruplets - will get sets of all four cards (e) can extend to two entire decks 3. non-messing-up theorem (a) shuffle (b) deal cards face up to form a rectangle (c) sort each row in increasing order (d) sort each column in increasing order (e) notice that rows are still ordered (f) why? 4. dilution (a) dived deck into red and black halves (b) take n cards from black half and put into red (c) shuffle red half (d) 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 cards? 5. Monty Hall (a) guess which of three cards has the ace (b) reveal which non-guessed card does not have an ace (c) what is the best 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+a + 1 + 7 3 + 1 3 2 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 3

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