Surely such a list must be available, but damned if I’ve been able to Google one up.

Using the frequencies from this page, 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.

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.

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.

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 Jyyyy, 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!

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.

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.)

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.

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.

When I saw this, I almost poked Reply to apologize and mutter something about Alzheimer’s ! On reconsideration, however, shouldn’t your 4[sup]52[/sup] actually be 4[sup]13[/sup] ? This times 13 yields about 850 million, I think, so my figure was an underestimate but not that dreadful.

Got it.[spoiler]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. :)[/spoiler]