1. You Too Can be a Mathematician Magician John Bonomo Westminster College ////////////////////////

2. The Basic Trick Volunteer picks one of 15 cards (call this the “key” card) Cards dealt in three piles, volunteer identifies “key” pile Pick up piles with key pile in the middle After three passes, key card is in the middle of the deck

3. Two Modifications Place key card in any location of the deck Allow users to pick up the decks

4. Analysis 0 3 6 9 12 1 4 7 10 13 2 5 8 11 14 ← Row 0 ← Row 1 ← Row 2 ← Row 3 ← Row 4

5. f(p) = 5 + p/3 Let p = position prior to deal Then new position is given by

6. p 3 = f(p 2 ) Let p 0 = initial position; p 1, p 2, p 3, positions after deals 1, 2 and 3 = f(f(p 1 ))= f(f(f(p 0 )))

7. p 3 = Let p 0 = initial position; p 1, p 2, p 3, positions after deals 1, 2 and 3 7 + 6 + p 0 27

8. Since 0 ≤ p 0 ≤ 14, we have p 3 = 7 + 6 + p 0 27 p 3 = 7

9. = 5 + 2 + 6 + p 0 27 p 3 = 7 + 6 + p 0 27 base offset

10. Generalized position function: Where i = 0 (bottom pile), 1 (middle pile), 2 (top pile) f i (p) = 5i + p/3

11. 0 i 1 1 2 0 i21i21 2 12 + p 0 4 + 27 6 + p 0 2 + 27 24 + p 0 3 + 27 18 + p 0 1 + 27 21 + p 0 2 + 27 15 + p 0 27 3 + p 0 1 + 27 9 + p 0 3 + 27 p 0 27 0 ≤ p 0 ≤ 14

12. 0 i 1 1 2 0 i21i21 2 24 + p 0 3 + 27 18 + p 0 1 + 27 21 + p 0 2 + 27 15 + p 0 27 0 ≤ p 0 ≤ 14 0 1 2 3 4

13. What’s your favorite number? Nine

14. 8 = 9 - 1 8 = 5 + 3 i 1 =0, i 2 =2, i 3 =1 Bottom, top, middle

15. 9 = 8 - 1 8 = 5 + 3 i 1 =0, i 2 =2, i 3 =1 Bottom, top, middle Why is your face so sweaty? And pale? Zzzzzzzzz…

16. 0 i 1 1 2 0 i21i21 2 24 + p 0 3 + 27 18 + p 0 1 + 27 21 + p 0 2 + 27 15 + p 0 27 0 ≤ p 0 ≤ 14 0 1 2 3 4

17. 0 i 1 1 2 0 i21i21 2 24 + p 0 3 + 27 18 + p 0 1 + 27 21 + p 0 2 + 27 0 ≤ p 0 ≤ 14 0 1 2 3 4 0, 0 ≤ p 0 ≤ 11 1, 12 ≤ p 0 ≤ 14

18. 0 i 1 1 2 0 i21i21 2 0 ≤ p 0 ≤ 14 0 1 2 3 4 0, 0 ≤ p 0 ≤ 11 1, 12 ≤ p 0 ≤ 14 1, 0 ≤ p 0 ≤ 8 2, 9 ≤ p 0 ≤ 14 2, 0 ≤ p 0 ≤ 5 3, 6 ≤ p 0 ≤ 14 3, 0 ≤ p 0 ≤ 2 4, 3 ≤ p 0 ≤ 14

19. 0 i 1 1 2 0 i21i21 2 0 ≤ p 0 ≤ 14 0: 100% 1: 100% 2: 100% 3: 100% 4: 100% 0, 0 ≤ p 0 ≤ 11 1, 12 ≤ p 0 ≤ 14 1, 0 ≤ p 0 ≤ 8 2, 9 ≤ p 0 ≤ 14 2, 0 ≤ p 0 ≤ 5 3, 6 ≤ p 0 ≤ 14 3, 0 ≤ p 0 ≤ 2 4, 3 ≤ p 0 ≤ 14

20. 0 i 1 1 2 0 i21i21 2 0 ≤ p 0 ≤ 14 0: 100% 1: 100% 2: 100% 3: 100% 4: 100% 1, 0 ≤ p 0 ≤ 8 2, 9 ≤ p 0 ≤ 14 2, 0 ≤ p 0 ≤ 5 3, 6 ≤ p 0 ≤ 14 3, 0 ≤ p 0 ≤ 2 4, 3 ≤ p 0 ≤ 14 0: 80% 1: 20%

21. 0 i 1 1 2 0 i21i21 2 0 ≤ p 0 ≤ 14 0: 100% 1: 100% 2: 100% 3: 100% 4: 100% 0: 80% 1: 20% 1: 60% 2: 40% 3: 60% 3: 20% 4: 80%

22. Always pick the “best” card 87% chance of selecting key card on first pick 100% chance of selecting key card on second pick (if necessary)

23. Generalize “Any Position” Trick n piles of m cards each still use only three deals Two questions: – What values of n and m work? – How do we determine i 1, i 2 and i 3 ?

24. m ≤ n 2 + gcd(n 2,m) 2 n (piles) m (cards in pile) 31,…,6,9 51,…,13,15,25 61,…,18,20,24,36 m ≤ n 2 + 1 2 (n,m relatively prime) Valid n,m pairs

25. Determine i 1,i 2 and i 3 for a given location L Let s = (L mod m) n 2 m Then i 1 = s mod n i 2 = s/n i 3 = L/m

26. Example: n=5, m=11 L = 40-1 = 39 i 1 = 14 mod 5 = 4 i 2 = 14/5 = 2 i 3 = 39/11 = 3 s = (39 mod 11) 5 2 11 = 14