The Gilbreath Principle
The Gilbreath shuffle:
Cut the deck at any point and reverse the order of one pack then riffle shuffle the two packs together
The Gilbreath 1st principle:
Set a deck alternating red and black and do a Gilbreath shuffle. The top two cards will always be
one red and one black
The Gilbreath 2nd principle:
Set a deck in any constantly repeated pattern of length m. After a Gilbreath-shuffle the top m cards
will always be a complete set of the set up.
The Gilbreath Ultimate principle:
For a deck of N cards ordered 1,..,N record a new order π(i) = the new position of the i-th card after
a Gilbreath-shuffle with i from [1,...,N]
we call π a Gilbreath permutation
For a permutation π of [1, 2, 3, … , N]the following properties are equivalent:
(1) π is a Gilbreath permutation.
(2) For each j, the top j cards [ π(1), π(2), … , π(j)] are distinct modulo j
(3) For each j and k with kj ≤ N the j cards [π((k-1)j+1), π((k-1)j+2), … , π(kj)]are
distinct modulo j
(4) For each j, the top j cards are consecutive in the original order (1, 2, … , N)
Example: (1 2 3 4 5 6 7 8 9 10) (5 6 7 8 9 10) (4 3 2 1)
(4 5 6 3 7 2 8 9 1 10)
The Mandelbrot Set
Set of all complex numbers c for which the sequence (z n ) with zn+1 = zn + c doesn’t diverge with
z0 =0
The Mandelbrot set has 2 dimensions: The real part of c (x-axis) and the imaginary party (y-axis).
This gives us
The connection between Mandelbrot set and shuffling cards:
Only the real part (the x-axis) matter
We call c a period n point if the (z n ) from the Mandelbrot set become 0 after n steps. Get c by
putting
zn=0 for a certain n. (there will always be a real solution since the degree is odd after excluding
one c)
Consider a period n 0point of a certain length n, call it c.
Example: for the period 4 point c = -1.3107
z0 = 0
z1 = (z0 ² + c) = -1.3107
z2 = (z1 ² + c) = 0,4072
z3 = (z2 ² + c) = -1,1448
z4 = (z3 ² + c) = 0
Write down the sequence of this c (starting with 0) and write above this a line the numbers 1, … , n
by putting 1 above the smallest number, 2 above the second smallest and so on, call this line (L)
In the example:
3
0
1
-1.3107
4
0,4072
2
-1,1448
This gives a “code” for a Gilbreath permutation.
Decoding:
Start with the 1, now always go one to the left and get another line, write this line under (L)
In the example we get
In
order:
3 1 4 2 (L)
1324
1234
3421
You will get a cyclic permutation. This method will always give a Gilbreath permutation as a result.
Sources:
 Persi Diaconis and Ron Graham: Magical Mathematics: The Mathematical Ideas that
Animate Greath Magic Tricks
 Nicolaas Govert de Bruijn: A Riffle Shuffle Card trick and its Relation to the Quasicrystal
Theory
 http://en.wikipedia.org/wiki/Mandelbrot_set