# Perfect Shuffles of Playing Cards

When a deck of playing cards is shuffled perfectly, the result is not random. A perfect shuffle places the cards in a mathematically precise order. As a result, when the most common version of a perfect shuffle is repeated eight times, the deck returns to its original state.

### Contents

#### A Deck of Playing Cards

Fresh out of its package, a deck of playing cards always has the Ace of Spades first, followed by the rest of the spades. The hearts, in order, are next, followed by the clubs (shown here in green), and finally the diamonds (blue). The King of Diamonds is last. Here is a view of a fresh deck.

v = 1:52; deck_view(v) title(0)

#### Faro Shuffles

A perfect shuffle is also known as a *faro shuffle*, because it is used frequently in the classic card game, Faro. The deck is cut into two packs, each containing exactly half of the cards. Then the two packs are merged by precisely interleaving the cards.

There are two variants. In an *out-faro* shuffle, the merger is started with the top card from the first half and eventually completed with the last card from the second half. So with an out-faro shuffle of a fresh deck, the Ace of Spades remains on top and the King of Diamonds remains on the bottom. An *in-faro* shuffle is started with the second half. So the Ace of Spades and the King of Diamonds are inserted in the second and next to last positions respectively.

#### Out-Faro

A shuffle is a permutation of the elements of a vector representing the deck. Here is the index vector that produces an out-faro shuffle. A rectangular matrix is created with `1:26` in the first row and `27:52` in the second. So its first column is `[1;27]`, its second column is `[2;28]`, and so on. The `reshape` operation, which is carried out by columns, then generates the desired index vector.

pout = reshape([1:26; 27:52],[1,52])

pout = Columns 1 through 13 1 27 2 28 3 29 4 30 5 31 6 32 7 Columns 14 through 26 33 8 34 9 35 10 36 11 37 12 38 13 39 Columns 27 through 39 14 40 15 41 16 42 17 43 18 44 19 45 20 Columns 40 through 52 46 21 47 22 48 23 49 24 50 25 51 26 52

Here is the result of two successive out-faro shuffles of a fresh deck.

v = 1:52; v = v(pout); deck_view(v) title(1)

v = v(pout); deck_view(v) title(2)

The four Aces have moved to the front and the four Kings are bringing up the rear.

#### It takes only 8 Out-Faro shuffles.

Let's start over with a fresh deck, apply the `pout` permutation eight times, and capture the results in an animated gif. The deck is returned to its original state in just 8 steps.

v = 1:52; deck_view(v) title(0) for t = 1:8 v = v(pout) deck_view(v) title(t) end

#### In-Faro

Here is the indexing vector for the in-faro suffle.

pin = reshape([27:52; 1:26],[1,52])

pin = Columns 1 through 13 27 1 28 2 29 3 30 4 31 5 32 6 33 Columns 14 through 26 7 34 8 35 9 36 10 37 11 38 12 39 13 Columns 27 through 39 40 14 41 15 42 16 43 17 44 18 45 19 46 Columns 40 through 52 20 47 21 48 22 49 23 50 24 51 25 52 26

It takes 52 in-faro shuffles to get back to the original state.

#### Permutation Matrices

Both of the faro shuffles can be represented by permutation matrices. Can you see how their `spy` plots differ?

```
I = eye(52);
close
Pout = I(pout,:);
spy(Pout)
title('Pout')
```

```
Pin = I(pin,:);
spy(Pin)
title('Pin')
```

It turns out that the smallest value of `t` for which the matrix power `P^t` is equal to the identity matrix is `t = 8` for `P = Pout` and `t = 52` for `P = Pin`.

#### Eigenvalues.

All of this is explained by *eigenvalues*. The matrix `Pout` has order 52, but only 8 distinct eigenvalues, namely the 8-th roots of unity.

plot(eig(Pout),'o') title('eig(Pout)') axis(1.25*[-1 1 -1 1]) axis('square')

On the other hand, `Pin` has 52 distinct eigenvalues, the 52-nd roots of unity.

plot(eig(Pin),'o') title('eig(Pin)') axis(1.25*[-1 1 -1 1]) axis('square')

#### Multiplicities

The eigenvalues of `Pout` occur with different multiplicites. Can you explain where these counts come from?

e = eig(Pout); z = exp(pi/4*1i) count = zeros(1,8); for k = 0:7 count(k+1) = sum(abs(e-z^k)<52*eps); end disp(' ') disp('multiplicites = ') disp(' 1 z i z^3 -1 z^5 -i z^7') disp(count)

z = 0.7071 + 0.7071i multiplicites = 1 z i z^3 -1 z^5 -i z^7 9 6 6 6 7 6 6 6

#### Deck_view.

Here is the listing of `deck_view`.

```
type deck_view
```

function deck_view(v) % deck_view(v) displays the card deck v. % Prepare the figure window. f = clf; f.Position = [200 300 600 150]; f.NumberTitle = 'off'; f.ToolBar = 'none'; f.MenuBar = 'none'; f.PaperPositionMode = 'auto'; % Prepare the axes. ax = axes; ax.Position = [0 0 1 .88]; ax.XLim = [-1 54]; ax.YLim = [-1 15]; ax.XTick = ''; ax.YTick = ''; ax.Box = 'on'; % Colors for spades, hearts, clubs, diamonds. shcd = [0 0 0; 1 0 0; 0 .5 0; 0 0 .5]; % Plot one text character per card. pips = 'A23456789TJQK'; for i = 1:52 j = mod(v(i)-1,13)+1; k = floor((v(i)-1)/13)+1; t = text(i,j,pips(j)); t.FontSize = 10; t.Color = shcd(k,:); end end

#### An informative video.

Kevin Houston is a mathematician at the University of Leeds in the UK who can also do perfect shuffles. Here is a link to his YouTube video. Perfect Shuffle.

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