A New Way to Look at Poker Hands

An interesting way for getting an intuitive sense for why certain hands are rare in poker, I've laid out all the cards in a standard deck in a grid: 

Array of Playing Cards

 

 

When I first learned to play poker, it took a while before I could get an intuitive understanding of the relationships between the various poker hands. Calculating their likelihood definitely helped get a handle on how many ways for a specific hand to actually occur there were.

 

Nonetheless, I think laying the hands out in a grid like this would have given me an intuitive understanding of the game much more quickly. 

Poker Hands shown geometrically

So the lowest poker hand, high card, is the most common happening 50% of the time, but will rarely be the winning hand:

High Card

 

The next highest hand is a single pair which has a geometric interpretation of one column with two dots in it: 

Pair

 

Two pair has a similar flair to it, except this one has 2 vertical lines:

 

Three of a kind is similar in spirit, but is much rarer than the two preceding hands happening once every 50 hands: 

3 of a kind

 

For a straight, we need 5 cards that are in order without any gaps and can look kind of like a nice scatter plot: 

Straight

 

A flush requires all 5 cards in the hand to be in a single horizontal row, but there can gaps between them:

Flush

 

A full house requires that all points be split into two lines of 3 and 2 points each:

 

Full House

 

Four of a Kind requires a vertical line that spans the entire grid with an extra point tucked away somewhere: 

4 of a kind

 

A straight flush is one of the tidiest hands as it requires all cards to be colinear on the same row and be immediately adjacent to each other:

Straight Flush

 

The best hand that you can get is a subset of all the straight flushes that you can get. It's just a straight flush all the way against the right side of the grid with the 5 highest cards: 

Royal Flush

 

Extending Poker

Looking at poker this way made me realize that there are some interesting hands that we could add to the game. 

Rectangle

This hand is formed when you have 4 points that are colinear in both orthogonal axes (form a rectangle): 

Rectangle Hand

 

Flower

This one seems like it would be complicated to spot in the wild, but in reality it's a lot like a full house. You need a Three of a Kind and the other two cards need to be the same suit as one of your 3oK cards, and just before and after it. So, maybe it is a little complicated to spot, but it certainly is an interesting idea. 

Flower Hand

 

Want More?

If you found this interesting, you'll probably love the 3D reconstruction we did of Walter White's cash pile in order to count it.

Also, we've compiled all these images into an infographic.

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