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Cribbage
Cribbage is an interesting (and old) card game. It probably dates back to 17th Century England. I’ll not duplicate the rules here, but sufficient to say they are a little different from traditional trick and trump based games.
The scoring happens in rounds, and various combinations/patterns of played cards are given different scores. If you are not familiar with the game of Cribbage, this article might not make a lot of sense, but the basic premise of the scoring is based on groupings in sets of five cards. A player has four cards in his/her hand, and there is a common card, which is “turned up” to make five.
Image: Kevin Doncaster
Scoring
Here are the (interesting) rules for scoring a set of cards in Cribbage. Points are combined.
If all four cards in the players hand are the same suit, this is a called a flush and scores 4 points. If the turned up card is also the same suit, the flush scores 5 points. (No points are scored for a flush of four cards that comprises three in the hand with the turned up card being the same suit).
Every pair of cards scores 2 points. If the hand contains three of a kind, this is called a pair royal and scores 6 points (as there are three combinations of pairs: AB, AC, BC). If the hand contains quads, this is called a double pair royal and this scores 12 points (representing six distinct combination of pairs: AB, AC, AD, BC, BD, CD).
Any combination of cards that add up fifteen pips, scores 2 points (in counting pips, all picture cards count as ten, and the Ace always counts as one). The fifteen count can be made using two, three, four, or all five of the cards. Each distinct combination of cards that add up to fifteen score two points.
A run of three consecutive cards scores 3 points. A run of four cards scores 4 points, and all five cards in consecutive order scores 5 points. Aces always count as one, and runs cannot go around the clock. Every distinct run is scored.
Finally, if the player holds a Jack which is the same suit as the turned up card, this is called his nob, and scores the player an additional 1 point.
Count all the hands
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What is the lowest possible scoring cribbage hand? What is the highest possible scoring hand? What is the distribution of the points of hands? One way to determine these is with a little brute force. We can simply calculate all possible combinations of hands, score these, and tabulate them. Sometimes brute force is the right tool for the job, especially in this case where the scoring is a superposition of individual, disparate, rules. Writing code to score the hands was also a fun little exercise. |
There are 52C4 combinations of four cards that the player can hold in the hand = 270,725
There are 48 possible remaining different cards that the turn up could be.
All possible hands: 270,725 × 48 = 12,994,800
The is a very manageable number of combinations for brute force. The code I wrote iterated through each of these combinations. I represented each card with an integer 0-51 and used modulus 13 arithmetic to decode the value and suit. A scoring function calculated the score of each hand based on the rules above and updated a frequency table for each score.
Results
The lowest possible score is 0 points. The highest possible score is 29 points; this is also the rarest occurring hand, makable only four ways. The most popular score is 4 points, achievable in 2,855,676 different ways (out of 12,994,800) or around 21.976%
Here are the results in tabular form:
| Score | Frequency | Percentage | Cumulative |
|---|---|---|---|
| 0 | 1,009,008 | 7.764706% | 7.764706% |
| 1 | 99,792 | 0.767938% | 8.532644% |
| 2 | 2,813,796 | 21.653246% | 30.185890% |
| 3 | 505,008 | 3.886231% | 34.072121% |
| 4 | 2,855,676 | 21.975529% | 56.047650% |
| 5 | 697,508 | 5.367593% | 61.415243% |
| 6 | 1,800,268 | 13.853757% | 75.269000% |
| 7 | 751,324 | 5.781728% | 81.050728% |
| 8 | 1,137,236 | 8.751470% | 89.802198% |
| 9 | 361,224 | 2.779758% | 92.581956% |
| 10 | 388,740 | 2.991504% | 95.573460% |
| 11 | 51,680 | 0.397698% | 95.971158% |
| 12 | 317,340 | 2.442054% | 98.413211% |
| 13 | 19,656 | 0.151261% | 98.564472% |
| 14 | 90,100 | 0.693354% | 99.257826% |
| 15 | 9,168 | 0.070551% | 99.328378% |
| 16 | 58,248 | 0.448241% | 99.776618% |
| 17 | 11,196 | 0.086158% | 99.862776% |
| 18 | 2,708 | 0.020839% | 99.883615% |
| 19 | 0 | 0.000000% | 99.883615% |
| 20 | 8,068 | 0.062086% | 99.945701% |
| 21 | 2,496 | 0.019208% | 99.964909% |
| 22 | 444 | 0.003417% | 99.968326% |
| 23 | 356 | 0.002740% | 99.971065% |
| 24 | 3,680 | 0.028319% | 99.999384% |
| 25 | 0 | 0.000000% | 99.999384% |
| 26 | 0 | 0.000000% | 99.999384% |
| 27 | 0 | 0.000000% | 99.999384% |
| 28 | 76 | 0.000585% | 99.999969% |
| 29 | 4 | 0.000031% | 100.000000% |
| TOTAL | 12,994,800 |
As you can see, it's not possible to make the score 19 points. It's an impossible score. Because of this, Cribbage vernacular refers to hands that have zero points as “nineteen point hands” or just “nineteen hands”.
(It's also not possible to make hands scoring 25, 26, or 27 points).
Of course, this exercise is a mathematical exercise in enumerating all the possible scoring combinations. In practice, when playing the game, a player is dealt six cards and has some control over the discards made and this adjusts the odds. (It's not possible to model this simply, however, as the discards a player may wish to make are based on situational strategy. Are you trying to simply maximize your personal score? Are you attempting to minimize the scoring potential in a crib for your opponent? Are you taking a gamble on what the turn up card will be? Are you trying to minimize your regret by spreading points over both your hand and your crib? …)
Below is the score data in histogram form:
And as a cumulative percentage of hands scoring at least that number of points:
Cribbage in the English Language
As mentioned earlier, the game of Cribbage is over 500 years old. It is mentioned in Charles Dicken's The Old Curiosity Shop.
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Tabulating of the scores in a game of Cribbage is typically performed on a cribbage board, instead of pen and paper. Pegs are inserted into drilled holes and advanced as points are earned (this is helpful, as scoring occurs frequently and rapidly). If the scores are tied, then the players are at "level pegging", which is a phrase used today in everyday speech to refer to when opponents are equal. Similarly if a helpful card is revealed in the turn-up, it's a "Turn up for the books", referring to an event that turned an, otherwise, not pleasant situation into something not so bad after all. (In cribbage, when a poor hand is converted into a nice scoring hand when a helpful turn card is revealed). |
The appearance of the correct Jack in a hand "his nobs", or sometimes "his nibs" also floated into vernacular in the region I grew up. My father often used to refer to a person as "his nibs", when referring, in jest, to an important person. e.g. "Well, we're all ready to go, so as soon as 'his nibs' finishes his breakfast we're off!"
Submarines
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Cribbage boards are available in an almost unlimited number of designs from the simple functional kind, to beautiful, elegant and sophisticated designs. I've seen boards shaped like ships, wheels, fish, flowers, canoes, trains, moustaches … An interesting piece of trivia is the naval tradition among submariners that the oldest submarine in the fleet be in possession of the cribbage board that belonged to Medal of Honor recipient and World War II prisoner of war Rear Adm. Richard H. "Dick" O'Kane. It's currently on-board the fast attack submarine, USS Bremerton (SSN 698). |
Highest Hands
Here's one example of the top scoring 29 point hand. There are 12 points achieved for the double pair royal (all four Fives), then eight different ways to make fifteen points (Jack with each of the four Fives, then the four different ways the three Fives can be combined to make fifteen). Finally, "his nibs" gives the extra point as it matches the suit of the turn-up card.
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29 points |
As there are only four different Jacks, there are only four possible ways to make a 29 point hand. It's such a rare occurrence that it's celebrated in the Cribbage community (just like a Royal Flush in Poker), and some Cribbage boards are shaped like the digits 29 in homage to this magic score.
28 points can be achieved by replacing the Jack with any ten-point card (any of the court cards plus or the ten), plus it's not important if this card appears in the hand or the turn up. Here's one possible example of the 76 ways this can be achieved.
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28 points |
There are no combinations of cards that can score 27, 26, or 25 points. There are 3,680 ways to score 24 points. There are a couple of different ways to achieve this. The first is with quads and singleton, such that two of the quads and the singleton total fifteen for example: 33339, 44447, 66663, 7777A. And other way is where there are quadruple runs of three, two pairs, and combinations of these add up to fifteen: 45566, 44566, 44556, 67788, 77889.
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24 points |
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24 points |
Scores of 22 and 23 are achievable through a pair royal of Fives and a pair of tens counts to give seven lots of fifteen. (To obtain 23 instead of 22 requires "his nibs").
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23 points |
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22 points |
Similarly 20 and 21 point hands differ only by the inclusion of "his nibs". Again there are multiple approaches. A double pair royal with a singleton such that three of quads and the singleton add up to fifteen: 22229, 33336, 44443. A double pair royal of Fives with the remaining card not contributing to the score: 5555A, 55552, 55553, 55554, 55556, 55557, 55558, 55559. A double pair royal and and singleton that make make fifteen: 66669, 77778, 88887, 99996, TTTT5, JJJJ5 (without the four that give the extra nibs points), QQQQ5, KKKK5. Using a pair royal of Fives, there are the hands: 555TJ, 555TQ, 555TK, 555JQ, 555JK, 555QK (where the ten count cards are different, and the Jack is not "his nibs"
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20 points |
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20 points |
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20 points |
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20 points |
Next there are the class of cards using runs. First the quadruple runs of three: 33445, 33455, 66778, 6788, 78899, 77899. It's also possible to have a double run of four with a four point flush: 34566, 44567, 66789, 67889, 67899. Finally, a double run of three with four point flush: A6778, 26778, A6788, 45669, 4556T, 4556J, 4556Q, 4556K, 5TTJQ, 5JJQK.
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20 points |
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20 points |
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20 points |
Trivia
If a player has four Aces in his hand, there is no possible turn-up card that could add any points to the (12 points) already obtained from the double pair royal. This is the only hand that cannot be improved by any turn card.
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12 points |
The highest possible scoring hand that has a five point flush is 18 points. In addition to the flush, it features "his nibs", a run of four, and four ways to make fifteen.
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18 points |
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