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View Full Version : A complete list of all 20+ point cribbage hands?


RTFirefly
09-07-2010, 12:11 PM
Surely such a list must be available, but damned if I've been able to Google one up.

Using the frequencies from this page (http://en.wikipedia.org/wiki/Cribbage_%28statistics%29), I'm confident that I've identified all the cribbage hands with 22 or more points, and ~97% (by frequency) of those with 20 or 21 points. But I'm clearly missing a few classes of hands.

Can anyone link to such a list? If not, I'll post what I've got, and we'll fill in the holes.

septimus
09-07-2010, 05:48 PM
8068 ways to score 20.
376) Double Pair Royal (20 ways each)
92222 63333 34444 A5555 25555 35555 45555 65555 75555 85555 95555 96666 87777 78888 69999 5TTTT 5QQQQ 5KKKK 5JJJJ (less 4 ways with nobs)

1404) Pair plus Pair Royal (120 ways each)
99333 77444 33666 99666 AA777 88777 77888 66999 55TTT 55QQQ 55KKK 55JJJ (less 36 ways with nobs)

1728) Pair Royal (320 ways each)
TQ555 TK555 QK555 JT555 (less 64 ways with nobs) JQ555 (less 64 ways with nobs) JK555 (less 64 ways with nobs)

4320) Quadruple Run (720 ways each)
33445 33455 66778 66788 78899 77899

72) Double Run of Four plus Four-flush (12 ways each)
34566 44567 66789 67789 67889 67899

168) Double Run plus Four-flush (12 ways each)
A6778 26778 A6788 45669 4556T (4 similar cases) 5TTJQ (6 similar cases)

2496 ways to score 21.
232) Ways shown under 20, but with nobs

24) Run and Flush (12 ways each)
55TJQ 55JQK

2240) Triple Run (320 ways each)
33345 44456 45666 67888 77789 67778 78889

444 ways to score 22.
444) Pair plus Pair Royal (120 ways each)
555TT 555JJ 555QQ 555KK (less 36 ways with nobs)

356 ways to score 23.
36) Ways shown under 22, but with nobs

320) Triple Run (320 ways each)
45556

3680 ways to score 24.
80) Double Pair Royal (20 ways each)
93333 74444 36666 A7777

3600) Quadruple Run (720 ways each)
44556 44566 45566 67788 77889

80 ways to score 28 or 29.
80) Double Pair Royal (20 ways each)
T5555 J5555 Q5555 K5555

RTFirefly
09-08-2010, 07:44 AM
It was the flush combinations that I was missing! I also undercounted (by 12) the number of hands that got their 21st point with his nobs, but since I missed the assorted 4-flush hands for 20 or 21 points, I never got to the point where that error was stumping me.

Thanks, Septimus!

septimus
09-10-2010, 04:20 AM
Cribbage can lead to fun puzzles. Here's one.

Suppose the rules of cribbage are left unchanged except that the sum which scores 2 points is changed from 15 to something else. Is there some "something else" that would allow a score higher than 29? Or also allow a score of 29?

My nine-year old son, Primus FitzSeptimus, plays cribbage fairly well. And he's been beating me as often as not at Casino, a "child's" card game which probably involves more skill than cribbage.

Dead Cat
09-10-2010, 06:28 AM
I'm not very experienced at cribbage, but I'll give this puzzle a go. Here's my working so far:

Firstly, I thought you would have to assume that "something else" > 10, otherwise the problem is too easy, but actually it doesn't really make any difference. Playing about with x = 12 and a hand of 84444 yields 28, but no more. Similarly x = 6 and 42222. So I think you need "his nob" to make 29. The problem then is that for JYYY y, J + y = 3y = x only when y = 5, and x must therefore be 15. I can't see another way of doing it, so I think 28 is the maximum score unless x = 15. A hint would be appreciated at this point if I am wrong!

MindWanderer
09-10-2010, 06:45 AM
Another interesting puzzle is to see what the highest score is, and what that best hand is for more than 5 cards.

I calculated it out once, out of curiousity.

I believe in the 6-10 card range, the 456 sequence dominates, and then a bit later, A2345 is where the best scores are, but the best scores go up VERY quickly with number of cards.

For 6 cards: "445566" is best, I think, with 46 points (8 possibilities for 15s for 16, an octuple run for 24, and 3 pair.). 5555JJ with suits working out for nobs is 39 points.

7 cards: "4455566" gives 72 points, and I think that's the best. 13 fifteens (12 from 456, 1 from 555), 12 runs, 2 pair and a triple.

8 cards: "44555566" gives 104 points.

The scores you can get keep going up fast from there.

septimus
09-10-2010, 10:37 AM
I can't see another way of doing it, so I think 28 is the maximum score unless x = 15. A hint would be appreciated at this point if I am wrong!

It does seem remarkable that the best number, 15, is the number chosen in the rules.

But there are two substitutions for 15 which would also yield a possible 29 top score. No further hint, except to note that when a cribbage player discovers the numbers, they may elicit bemusement or a chuckle....

(PS: My son asked me what the 52-card cribbage hand would count to. I think I told him 700 million, give or take.)

Hoopy Frood
09-10-2010, 12:47 PM
What about the sum of 20?

J(1) J(2) Q(1) Q(2) and a cut card of the K of one of the Jack suits.

You get J(1)-J(2), J(1)-Q(1), J(1)-Q(2), J(1)-K, J(2)-Q(1), J(2)-Q(2), J(2)-K, Q(1)-Q(2), Q(1)-K, and Q(2)-K for your 2-point sums.

That gives you a score of 20 for your sums.

Plus, you get a quadruple run for 12.

Plus you get nobs for another 1.

So voila, 33.

ETA: You can also accomplish this with four Jacks in your hand and any other face card or ten as a cut card. There you get another 10 sets of 20 point sums, four of a kind for 12 and nobs for 1 more.

septimus
09-10-2010, 02:07 PM
What about the sum of 20?
... That gives you a score of 20 for your sums.
Plus, you get a quadruple run for 12.
Plus you get nobs for another 1.
So voila, 33.

16 for quadruple run (i.e. 4 for the two pairs); total is 37. There's another total beside 20 that works just like this.

Somehow, I overlooked these when I posted earlier. :smack: :smack:

Puzzle remains. 20 and another number allow 37 total. 15 and two other numbers allow 29 total. What are the other numbers?

Hoopy Frood
09-10-2010, 08:54 PM
What about the sum of 20?
... That gives you a score of 20 for your sums.
Plus, you get a quadruple run for 12.
Plus you get nobs for another 1.
So voila, 33.

16 for quadruple run (i.e. 4 for the two pairs); total is 37. There's another total beside 20 that works just like this.

That's right. Don't ask me why I forgot to add in the two pairs.

So yeah, 37.

And as far as the other sum total, it's 30.

J1,J2,Q1; J1,J2,Q2; J1,J2,K; J1,Q1,Q2; J1,Q1,K; J1,Q2,K; J2,Q1,Q2; J2,Q1,K; J2,Q2,K; Q1,Q2,K

Once again 10 sets of 2 points each for 20 for your sums.

Malacandra
09-11-2010, 02:23 AM
Amazing! Commentary: 30 and 20 both work the same because there are as many "2 from 5" combos as "3 from 5" - 5!/3!2! = 10 in both cases.

700 million looks like a dreadful underestimate for the 52-card hand though. There are 452 13-card runs, for a start... :D

septimus
09-11-2010, 04:54 AM
Amazing! Commentary: 30 and 20 both work the same because there are as many "2 from 5" combos as "3 from 5" - 5!/3!2! = 10 in both cases.

Stated differently, the 37-scoring hands total 50-count, so there will be exactly as many ways to count (50-30) as to count 30.

In ordinary cribbage, when I have a complicated hand with total count 22, for example, I find it easier to locate the 7-counts rather than the 15-counts. The numbers will be the same.


700 million looks like a dreadful underestimate for the 52-card hand though. There are 452 13-card runs, for a start... :D

When I saw this, I almost poked Reply to apologize and mutter something about Alzheimer's ! On reconsideration, however, shouldn't your 452 actually be 413 ? This times 13 yields about 850 million, I think, so my figure was an underestimate but not that dreadful. :D

RTFirefly
09-11-2010, 08:42 PM
But there are two substitutions for 15 which would also yield a possible 29 top score. No further hint, except to note that when a cribbage player discovers the numbers, they may elicit bemusement or a chuckle....Got it.Bemusement or a chuckle, huh? That hint practically screamed, "19!!!!" I'm not proud; I tried 19.

The hand is 9-9-10-10-J, or 9-9-10-J-J, with the right jack. You've got 6 different 19s, for 12 points, the quadruple run for another 16, and his nobs for the 29th point.

And once you get 19, even if you don't immediately realize that you already possess the other solution with the very same hands, reading the discussion in posts 10, 11, and 12 of this thread should do the trick. 29 is the other solution. :)

Malacandra
09-12-2010, 02:10 AM
Amazing! Commentary: 30 and 20 both work the same because there are as many "2 from 5" combos as "3 from 5" - 5!/3!2! = 10 in both cases.

Stated differently, the 37-scoring hands total 50-count, so there will be exactly as many ways to count (50-30) as to count 30.

In ordinary cribbage, when I have a complicated hand with total count 22, for example, I find it easier to locate the 7-counts rather than the 15-counts. The numbers will be the same.


700 million looks like a dreadful underestimate for the 52-card hand though. There are 452 13-card runs, for a start... :D

When I saw this, I almost poked Reply to apologize and mutter something about Alzheimer's ! On reconsideration, however, shouldn't your 452 actually be 413 ? This times 13 yields about 850 million, I think, so my figure was an underestimate but not that dreadful. :D

Aargh! :smack::smack::smack::smack::smack::smack::smack::smack:

Math exam anxiety starts now. :(

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