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Sevendeck Poker

Sevendeck Poker Rules Summary

Sevendeck Poker is played like regular poker, with a few adjustments. First, mere pairs are of no value. Multiples of more than two have value; the pairs contained in them don't "cancel". One consequence of this is that there's no Full House; a hand with three of one value and two of another is just Three of a Kind.

Second, there are no flushes. Instead, there is a "Stretch", which is five cards with consecutive suits. Suits wrap around, as suggested by the circle of suits on the box. So, for example, Diamond-Violet-Cherry-Fire-Lemon would be a Stretch.

When a hand is both a Stretch and a multiple, the higher hand takes precedence. There is no special hand category for Stretch-multiple combinations. (Pairs in a Stretch do not cancel; they still contribute their suit.) Straight Stretch is still a special hand, like the Straight Flush in regular Poker. For a Straight Stretch, the values and suits do not have to be in the same sequence.

Sevens can be low (zero) in a Straight. (Ones ("aces") cannot be high.)

The order of the hands, from highest to lowest, is:

Notice how High Card beats Two Pair and Pair. High Card is always a five-card hand, like a Straight or a Flush. (This might matter if you're playing a travesty like Guts where hands may be scored with fewer than five cards.)

When breaking ties in Two Pair or Pair, use the non-paired "kickers" first, then move on to the pairs as a last resort.


What follows is a math-geek-y discussion of where the above rules came from.

An Analysis of Poker with a Sevendeck

A Sevendeck is a deck of 49 cards, with seven suits and seven cards in each suit, numbered 1 through 7. Unlike a regular deck of cards, the suits fall into a well-defined sequence, based the number of angles in the symbol as well as the spectrum order of the colours.

One question that this deck raises is how the game of Poker would work on such a deck. The following is an attempt to answer that.

The most direct approach would be to simply use all the standard rules of Poker, unmodified. The odds of being dealt a five-card hand of a specific type are as follows:

HandFrequencyProbability
Five of a Kind 1470.000077
Straight Flush 280.000015
Four of a Kind 102900.005396
Full House 308700.016189
Flush 1190.000062
Straight 672000.035241
Three of a Kind 1800750.094434
Two Pair 3241350.169981
Pair 10084200.528831
High Card 2856000.149773

This is sort of like playing a variant of Poker with wild cards, where the probabiltites of different hands are no longer monotonically increasing. We could instead re-order the values of the hands to put them in probability order:

HandFrequencyProbability
Straight Flush 280.000015
Flush 1190.000062
Five of a Kind 1470.000077
Four of a Kind 102900.005396
Full House 308700.016189
Straight 672000.035241
Three of a Kind 1800750.094434
High Card 2856000.149773
Two Pair 3241350.169981
Pair 10084200.528831

This has the interesting consequence of making flushes the most valuable types of hands. However, it also makes them so rare that they are nearly irrelevant to game play. This happens because in a Sevendeck, a Flush is just like Five of a Kind, but with suits instead of numbers. The fact that the Straight Flush splits this small set of hands into two smaller subsets means that they are each more rare than Five of a Kind.

Also, notice that pairs are abundant. With only seven different numeric values, there is more than a 50% chance that any set of five will be a Pair. If all hands involving multiples are factored in, there is more than an 81% chance of getting a hand with at least two of a single value. This makes High Card more rare than Two Pair or Pair.

With the Sevendeck's rich supply of novel suits, it seems like a waste to make the suit-based hands so rare that they have basically no effect on the game. Therefore, we need a new kind of suit-based hand that has a better chance of coming up in a set of five cards.

Unlike in regular Poker, the Sevendeck has a well-defined relationship among its suits. We can use this relationship to define a new type of hand, called a Stretch, which consists of five cards with consecutive suits. The sequence of suits wraps around, as suggested by the circle of suits on the box and the spectral colouring of the suits. So, for example, Diamond-Violet-Cherry-Fire-Lemon would be a Stretch.

Unlike the Flush, a Stretch is relatively easy to obtain in a set of five cards. Also unlike the Flush, a Stretch is not mutually exclusive with a multiple-value hand like Pair or Full House. To facilitate computing odds, we can start with the "raw" frequences of all the possible types of hands:

HandFrequency
Five of a Kind Stretch 49
Five of a Kind 98
Four of a Kind Stretch 1470
Full House Stretch 2940
Straight Stretch 3360
Four of a Kind 8820
Stretch 14280
Three of a Kind Stretch 14700
Two Pair Stretch 22050
Full House 27930
Pair Stretch 58800
Straight 63868
Three of a Kind 165375
High Card 271439
Two Pair 302085
Pair 949620

One could simply attempt to play the game with these sixteen hands, but the sequence is weird and hard to remember. It would be nice to have a smaller set of distinct hands that have a simpler relationship.

A key aspect of the order of hands in Poker is that any hand that is more constrained than another is worth more. For example, a Full House beats Three of a Kind. The above sequence does not have this property. Stretch beats Three of a Kind Stretch, Two Pair Stretch, and Pair Stretch. As noted in the second "standard Poker" version above, High Card still beats Two Pair and Pair. However, this latter condition is kind of an interesting twist that seems worth keeping. After all, the goal is to produce a game that is similar enough to Poker to still resemble the game, yet different enough to be interesting.

However, the placement of Stretch in this sequence is awkward and calls for improvement. The position of Pair at the bottom of the list suggests a simple fix: "Pairs are worth nothing". We can further condense the list by declaring that for any hand that has the Stretch property and a multiple value, it acts as the higher valued hand---there is no special category for such a "combination" hand. We'll keep the Straight Stretch, though, because it parallels standard Poker well. When we condense the list of hands in this manner, we get the following simple sequence:

HandFrequencyProbability
Five of a Kind 1470.000077
Straight Stretch 33600.001762
Four of a Kind 102900.005396
Straight 638680.033493
Stretch 1127700.059138
Three of a Kind 1933050.101372
High Card 2714390.142347
Two Pair 3020850.158418
Pair 9496200.497996

Note how a Stretch is more likely than a Straight, which is the opposite of the relationship between Flush and Straight in regular Poker. This arises because there are more possible starting points for the sequence. With only one card value "wrapping around", a Straight can only start at four possible values. With total wraparound of suits, a Stretch has seven possible starting points. Therefore, there are more possible Stretch hands, even though a few are absorbed into Four of a Kind and Five of a Kind.

We still have the odd property that the less-constrained High Card hand is rarer than Two Pair or Pair, but this quirk gives the game a nicely different property than Standard Poker. If you like, think of High Card as a hand consisting of five different values which are not sequential.

One more twist to make the game more "Sevendeck-y" is to allow 7 to be low in a straight, instead of allowing 1 (the "Ace") to be high. This puts the specialness on the Seven, which seems appropriate for a deck built around that number.

An open question is whether Two Pair and Pair should be lumped together into a single group or not. What is clear is that when breaking ties within those two groups, the pairs should take lower precedence. So 2-2-3-3-7 beats 4-4-5-5-6, for example. If the kickers match, then the pairs can be used to break ties.