In considering the accomplishment of Phil Ivey at Crockfords Casino (see this post), it is worth considering the full breadth of the problem of ill-designed cards and edge sorting. Phil Ivey did not get lucky by finding an especially rare card in a far-away location that could be sorted. My personal survey of 650 casino decks showed that fully 70% of all cards contain asymmetries that allow them to be sorted. Poorly manufactured cards are ubiquitous. Because of this fact, sorting has been going on for a very long time in casinos. Abram Alexander mentions sorts in his book "Advanced Tactics in Casino Advantage Play," saying that the method has been around for "decades."

What follows is a sampling of some of the most common cards and some of the most common issues.

The first card represents the most commonly used card design.  Over 20% of all cards in use have this design. This card is also highly sortable.  Notice the difference in the size and shape of the diamonds on each side of the card. You will see half-diamonds on the right-hand side and very small fragments on the left-hand side. A similar issue occurs on the top and bottom of the card.

half-diamonds on the right-hand side and very small fragments on the left-hand side. A similar issue occurs on the top and bottom of the card


The following card uses the second most common design for its back. Over 15% of all cards in use have this design. In my opinion, of all the sortable cards, the asymmetry in the cut of this particular design is most clearly visible.

ard uses the second most common design for its back. Over 15% of all cards in use have this design


Here are a few more cards that share the issue of having asymmetric cuts along the left and right-hand edges of the card. Each is asymmetric along the top and bottom edges as well:

asymmetric cuts along the left and right-hand edges of the card

 

asymmetric along the top and bottom edges as well

 

asymmetric along the top and bottom edges as well


The following card would be a challenge to sort by considering the top and bottom edge, but the left and right edges are easily distinguished:

a card would be a challenge to sort by considering the top and bottom edge, but the left and right edges are easily distinguished


The following cards are cut well enough along the left and right edges, but the top and bottom edges are clearly distinguishable:

cards are cut well enough along the left and right edges, but the top and bottom edges are clearly distinguishable


Just because a card has a bleached border does not mean it is safe from sorting. Though rare, cases do occur when the diameters of the bleached boarders along two opposite sides are clearly different. In the following example, you can clearly see that the left hand border is much wider than the right hand border:

Back of  a card image - the left hand border is much wider than the right hand


In this example, the top border is much wider than the bottom border:

An image of a back of a card - the top border is much wider than the bottom border


Sometimes, the card looks entirely unsortable. However, complex designs can lead to differences in opposite corners of the card. This can also be used to sort the cards. In the following example, compare the upper right corner to the lower left corner:

An image of a back of a card - complex designs can lead to differences in opposite corners of the card


In very rare cases, the logos can be imprinted so that they are not placed symmetrically with respect to the underlying design. In the following card, notice there are half-ovals underneath the top logo (beneath the word "casino"), but there are full ovals above the bottom logo (above the word "casino").

An image of a back of a card - half-ovals underneath the top logo (beneath the word "casino"), but there are full ovals above the bottom logo (above the word "casino")


Sometimes, an asymmetric design is used in a key location on the card. This design by itself is sufficient to allow sorting of the card. Consider, for example, the five-pointed star at the center of this card:

An image of a back of a card - the five-pointed star at the center of this card


Finally, sometimes mistakes in design are made that defy logic. I leave this card as a puzzle to you:

An image of a back of a card - mistakes in design are made that defy logic


Even though these cards are all sortable, there is still one very simple tactic that will unilaterally defeat card sorting: include a turn in every shuffle procedure, for every game.

About the Author
By

received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson

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