1. You Too Can be a Mathematician Magician
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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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ← Row 0 ← Row 1 ← Row 21 2
← Row 0 3 4 5
← Row 1 6 7 8
← Row 2 9 10 11
← Row 3 12 13 14
← Row 4

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

6. Let p0 = initial position; p1, p2, p3, positions after deals 1, 2 and 3
p3 = f(p2)
= f(f(p1))
= f(f(f(p0)))

7. Let p0 = initial position; p1, p2, p3, positions after deals 1, 2 and 3
7 +
6 + p0 27
p3 =

8. 6 + p0
p3 =
7 + 27
Since 0 ≤ p0 ≤ 14, we have
p3 = 7

9. p3 =
7 +
6 + p0 27 =
6 + p0 27
base
offset

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

11. i2 1 2
i1 1 2 p0 27
18 + p0
1 + 27
9 + p0
3 + 27
15 + p0 27
6 + p0
2 + 27
24 + p0
3 + 27
3 + p0
1 + 27
21 + p0
2 + 27
12 + p0
4 + 27
0 ≤ p0 ≤ 14

12. i2 1 2
i1 1 2
18 + p0
1 + 27 3
15 + p0 27 2
24 + p0
3 + 27
21 + p0
2 + 27 1 4
0 ≤ p0 ≤ 14

13. What’s your favorite number?
Nine

14. 8 = 9 - 1
8 = 5 + 3
Nine
i1=0, i2=2, i3=1
Bottom, top, middle

15. 9 = 8 - 1
Why is your face so sweaty?
8 = 5 + 3
i1=0, i2=2, i3=1
Zzzzzzzzz…
Bottom, top, middle
And pale?

16. i2 1 2
i1 1 2
18 + p0
1 + 27 3
15 + p0 27 2
24 + p0
3 + 27
21 + p0
2 + 27 1 4
0 ≤ p0 ≤ 14

17. i2 1 2
i1 1 2
18 + p0
1 + 27 3
0, 0 ≤ p0 ≤ 11
1, 12 ≤ p0 ≤ 14 2
24 + p0
3 + 27
21 + p0
2 + 27 1 4
0 ≤ p0 ≤ 14

18. i2 1 2
i1 1 2
1, 0 ≤ p0 ≤ 8
2, 9 ≤ p0 ≤ 14 3
0, 0 ≤ p0 ≤ 11
1, 12 ≤ p0 ≤ 14 2
3, 0 ≤ p0 ≤ 2
4, 3 ≤ p0 ≤ 14
2, 0 ≤ p0 ≤ 5
3, 6 ≤ p0 ≤ 14 1 4
0 ≤ p0 ≤ 14

19. i2 1 2
i1 1 2
1, 0 ≤ p0 ≤ 8
2, 9 ≤ p0 ≤ 14
0: 100%
3: 100%
0, 0 ≤ p0 ≤ 11
1, 12 ≤ p0 ≤ 14
2: 100%
3, 0 ≤ p0 ≤ 2
4, 3 ≤ p0 ≤ 14
2, 0 ≤ p0 ≤ 5
3, 6 ≤ p0 ≤ 14
1: 100%
4: 100%
0 ≤ p0 ≤ 14

20. i2 1 2
i1 1 2
1, 0 ≤ p0 ≤ 8
2, 9 ≤ p0 ≤ 14
0: 100%
3: 100%
0: 80%
1: 20%
2: 100%
3, 0 ≤ p0 ≤ 2
4, 3 ≤ p0 ≤ 14
2, 0 ≤ p0 ≤ 5
3, 6 ≤ p0 ≤ 14
1: 100%
4: 100%
0 ≤ p0 ≤ 14

21. i2 1 2
i1 1 2
1: 60%
2: 40%
0: 100%
3: 100%
0: 80%
1: 20%
2: 100%
3: 20%
4: 80%
2: 40%
3: 60%
1: 100%
4: 100%
0 ≤ p0 ≤ 14

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 i1, i2 and i3?

24. Valid n,m pairs n (piles) m (cards in pile) 3 1,…,6,9 5 1,…,13,15,25 6
1,…,18,20,24,36
m ≤
n2 + gcd(n2,m) 2
m ≤
n2 + 1 2
(n,m relatively prime)

25. Determine i1,i2 and i3 for a given location L
Let s =
(L mod m) n2 m
Then i1 = s mod n
i2 = s/n
i3 = L/m

26. Example: n=5, m=11 L = 40-1 = 39 s = (39 mod 11) 52 11 = 14
i1 = 14 mod 5 = 4
i2 = 14/ = 2
i3 = 39/ = 3