How Math Explains the World.pdf

the shuff le that doesn’t change the position of any card—the “phantom
shuff le” that is sometimes performed by magicians or cardsharps. The
inverse of any shuff le is the shuff le that restores the cards to their origi-
nal position. For instance, we can use the above diagram to get a look at a
portion of the inverse to the perfect shuff le.

Starting

Position 1 2 3 4... 49 50 51 52

Ending

Position 1 27 2 28... 25 51 26 52

Again, using algebraic notation.

Starting Position (x) Ending Position

x is odd (x 1)/2

x is even 26 x/2

To see that this is indeed the inverse of the perfect shuff le, notice that if
a card starts out in position x, where 1 x26, the perfect shuff le puts
it in position 2x
1 (an odd number), so the inverse puts it in position
((2x
1) 1)/2x—back where it started. If a card starts in position x,
where 27x52, the perfect shuff le puts it in position 2x
52 (an even
number), so the inverse puts it in position 26(2x
52)/2 x—again,
back where it started. Similarly, one can show that if one performs the
inverse first and follows it with the perfect shuff le, every card returns to
its original position. Although it is not germane to the quintic problem,
performing eight perfect shuff les of a deck of fifty-two cards restores the
deck to its original order—if g denotes the perfect shuff le, this is written
g^8 1, and mathematicians say that g is an element of order 8. Showing
that shuff ling satisfies the associative law is not difficult—but it isn’t es-
pecially interesting, so I’ll skip the demonstration.
Notice that the perfect shuff le—and its inverse—leave the top card of
the deck unchanged. If we were to consider all the shuff les that leave the
top card of the deck unchanged, we would discover that they also form
a group—the product of any two such shuff les leaves the top card un-
changed, and the inverse of such a shuff le also leaves the top card un-
changed. A subset of a group that is itself a group is called a subgroup.
One way in which the group of all shuff les differs from the group of
nonzero real numbers is that the latter group is commutative—no matter
which order you multiply two numbers, the result is the same: for exam-

92 How Math Explains the World