Rules

David Steinbrenner from Wynnewood, Pennsylvania writes:

"Please clear up an ongoing rules 'discussion' I am having with my father and brother-in-law. We are avid cribbage fans and enjoy the game, and the fellowship the game brings, immensely. Upon reaching 31 exactly with the final last card, what is pegged? This may seem like an obvious question, but reading many sets of rules has not cleared it up. I believe three points should be pegged: two for 31 and one for last card. My cribbage mates believe only two should be pegged, not counting the one for last card. I hope my question is clear. I would appreciate your learned opinion. Thank you, and I enjoyed your Web site."

David, the rules for scoring a go or last card are probably the most confusing in cribbage. The answer to your question is: 31 is worth just two points, even if it comes on the last card played. I'm afraid your relatives are right!

Not every rule in cribbage is logical, but this one does have a rationale behind it, which you can grasp if you accept that a go and last card are basically the same thing. In effect, you're awarded one point when your opponent is unable to make a legal play, either because:

• he has no cards remaining that would keep the count within 31 (a go), or
• he has no cards remaining at all (last card)

If you play a card that makes the count 31, what you get is:

• one point for either go or last card, plus
• one "bonus" point for hitting 31 exactly

So the two points you peg for 31 already include the point for go or last card. I hope this explanation helps.

I'm glad you enjoy the game, and that it's a source of fellowship for you and your associates. This, along with a degree of mental conditioning, is the most important thing you can get out of cribbage!

Discarding

Klagett, from Upstate New York, submitted this via the Yahoo! Cribbage Message Board:

"I decided after my last game, when my opponent asked me if we were playing lowball, that I was going to have to learn how to play better cribbage.

OK, I got 2 4 5 8 10 J. It's my deal. I threw the 10-J. How would you have played it?"

First off, let me say that any question about discarding must take board position into consideration. Different scores often call for discarding the same hand in different ways (and as we'll soon see, 2-4-5-8-10-J is a good example of this). Without knowing what the score was when you got this hand, I can only give you general advice.

In most situations, an expert player will toss either 2-4 or 2-8 from this hand. Let's see how these two discards perform relative to each other, and relative to the other alternatives, including your own 10-J toss:

Keep:	Toss:		Average  hand:	Chambers  average:		Average crib:			Expected average:
						Hessel	Colvert	Ras	Hessel	Colvert	Ras
2-4-5-8	10-J		5.13	6.43		4.64	4.5	4.76	9.77	9.63	9.89
2-4-8-10	5-J		2.35	5.61		7.04	6.9	7.09	9.39	9.25	9.44
2-5-8-10	4-J		6.26	7.09		3.89	3.8	3.98	10.15	10.06	10.24
4-5-10-J	2-8		7.22	7.78		3.58	3.6	3.82	10.80	10.82	11.04
5-8-10-J	2-4		6.57	7.26		4.51	4.4	4.64	11.08	10.97	11.21

The most important number in evaluating discards as dealer is the expected average, which indicates the average combined value of your hand and crib after the cut. As you can see, the 2-4 and 2-8 tosses are clearly at the head of the pack, and either one is a good choice if you're playing on or playing cautious offense. Tossing 2-4 returns a higher expected average than tossing 2-8, although its lower Chambers average indicates that it carries more risk, since it places more of its scoring potential in the crib, which is subject to your opponent's balking efforts. Tossing 2-8 will get you about ¼ point less between hand and crib, but it leaves you with 4-5-10-J, which is a slightly better offensive pegging hand due to its two sets of touching cards. 4-5-10-J pegs one point more than 5-8-10-J if pone has four ten-cards. And you might even get lucky and trap pone's 6 for a 15-6 at the end of the play. If you're playing off however, definitely keep 5-8-10-J. The 8 gives you a comfortable reply to a low card lead, something that's hard to defend against holding 4-5-10-J.

Note that if you apply the addition method to this hand, you'll be correctly guided to the 5-8-10-J keep, since among the three keeps worth four points in the hand, this is the one with the highest average crib.

Of course as I mentioned earlier, board position trumps everything. If the score is 113*-108, for example, your best hand is probably 2-5-8-10, keeping four well-spaced cards for defense (a lone J is a defensive liability, so I prefer to keep the 10, even though the J would give me a safe play on an 8 lead). If the score is 118*-118, then keep 2-4-5-8, your best offensive pegging hand.

For more about discarding as dealer, have a look at the Discarding to your crib article. If you're not familiar with terms such as average hands, expected averages and Chambers averages, I suggest you review the series of Cribbage Forum articles on calculating discards, beginning with How to analyze discards, part 1.

Speaking of discarding, Life Master George "Ras" Rasmussen sent me this in October 2000:

"Mike, you have done a great job here in presenting material relative to discarding. Your format is enlightening and your explanations where offered seem logical. I like it very much. This is a great service to those who will take the time to review and apply the concepts to actual play. Gee, I am off to a great start on the new season with 209 master rating points after the first month. First time ever that I've won two Mains in two months. I intend to fully follow your guides to discarding. Thanks much for sharing."

Hey Ras, thanks for the kind words, and congratulations on your fast start to the 2000-2001 tournament season! Everyone who follows cribbage closely knows about your tremendous contributions to cribbage theory, and your role in popularizing the game here in North America.

Pegging

Several readers wrote in with pegging vignettes. A Yahoo! Games player named niner1 writes:

"I would like to say I was impressed with your info on cribbage. I just started at Yahoo! Games and went 20-1, so I'm a pretty good player myself. A lot of it was luck, but that one game I did lose was to a opponent that was very aggressive about pegging on my first card. What would you lead if you didn't have a pair, say from 2-8-9-J? I normally would lead a 2 realizing the opponent might throw down a 9 to try to make me hold my ten-card and play another low card. But if I pair the 9 I'm in big trouble if dealer has another 9."

Hey, congratulations on your fast start at Yahoo! Regarding your question: as you've apparently noticed, the problem with leading a 2 from 2-8-9-J is that you're trapped on a 9 or 10 reply. That's why the 8 is a better lead in most situations. If dealer takes a 15-2 with a 7, you can either take the run with your 9 (if you're playing on) or break with your 2 (if you're playing off). On a ten-card reply, you can either play your J (perhaps pairing dealer's J) and hope for a go, or else you can break with the 2, keeping the count under 21 and preventing dealer from getting a three-on-one on the second play series if he has nothing but high cards.

If you're playing desperation offense or desperation defense, however, then the 2 is a good lead. Say the score is 114-109*, and you've cut the right 10. You need to peg two points to win, and you can assume your opponent will be playing defense. Lead your 2. It could well catch an 8, 9 or even a 5 in reply (a 3, 4 or J would be nice too, but unlikely). If you can't score off dealer's first card, try to save your 8-9 for last, hoping to trap dealer into a run later. Now imagine the same hand and cut, but at 116-118*. This time you have enough points in your hand to go out, but dealer only needs to peg three to win. Since you can't afford to give up even a two-point score, you should lead your 2, the card dealer is statistically least likely to be able to score on.

A Zone player who goes by Madison_wi, asks:

"The score is 116*-116. What would you choose to be dealt if you know nothing about your opponent?"

Pone has a better than 80% chance of holding a hand worth at least five points after the cut, so I'm going to try to win by pegging out. The best hand I can imagine in this situation is A-A-A-8. I'd play the 8 on anything but an A lead, and hope to run the three As for a triple. If pone is nice enough to cut me a J, all I'll have to do is run two of the As to win. Another good hand to hold needing to peg five to win would be something like 4-5-6-7. This scores two points on anything but an A, 2 or 3 lead, and has lots of potential for pairs, 15s, and runs later in the pegging. If I can find just two two-point scores, then I'll win with last card — provided of course, that pone doesn't peg out first.

Hal Mueller, the author of HALSCRIB, offers the following:

"It was with some interest that I read your article on leading a 5. About five years ago when I was debugging my pegging algorithm, HALSCRIB led a 5 quite often. So I was not very careful in one game where I had A-4-J-Q and HAL led a 5. I played Q and HAL played a 9. I carelessly played a 4 and then the roof fell in. HAL pegged 31 for 8:

⁵  _Q (15-2)  ⁹  ₄  ^A  _A (30-2)  ^A (31-8)

I was so impressed with this play that I planned to use it in my next tournament. The very first hand of the first game of a tournament in Buffalo (at the Air Force base, the year of the Hale-Bopp comet) I happened to play Jim Bough, a veterinarian from Grand Rapids whose wife had purchased the DOS version of HALSCRIB years earlier. He didn't fall for the trap, but we had a good laugh about it after the hand was over. I still use the play (when playing offense) because most good players like yourself avoid playing a ten-card. In that same tournament, I held 6-6-7-7 as dealer (playing offense). My opponent led a J from 5-5-J-J. Needless to say, I pegged seven points after he played his other J. (When playing defense, I will lead a 5 from 5-5-x-x — another play I learned from HALSCRIB). I was the last qualifier and if I had known then what I know now, I may have won instead of finishing 2nd overall."

Hal, it must be particularly gratifying to learn a trick or two from your "baby". We'll all look forward to your continued work on HALSCRIB!

Incidentally, on the topic of leading a 5, George Rasmussen has the following to say in the November 2000 issue of Cribbage World:

"Have you ever faced the dilemma of holding 5-5-5-x? And it's your lead? Do you lead a 5 or the ten-card?

I have analyzed this situation, and it is far better to lead a 5!

If you lead your ten-card, the pegging may very well go:

¹⁰  ₄  ⁵  _3 (22-3)  ⁵  _4 (31-5)

Your opponent scores eight points. And you're stuck leading your last 5 to begin a new sequence. By leading a 5 [to begin with], you will probably be burned by only a 15-2, and a possible second 15-2, risking four points."

Personal database

Duane Stansbie from Toowoomba, Australia asks:

"Reading through your pages you mention keeping your own cribbage database. What do you record in it? The system I am trying to adopt is to record cards dealt, and cards discarded from both players. After the game I'm then trying to see what mistakes were made. You advocate having your database available during play. Do you have a database which says 'from this hand discard this, and lead this'?"

Duane,

My database has literally hundreds of records, since I use it to keep track of published hands analyzed by myself and other authors. Since this gets a bit unwieldy, I've flagged 100 or so of these records as being particularly important. These are the entries I review regularly to reinforce essential ideas. The others I look at only occasionally. I keep my database in an Excel spreadsheet, organized into pone and dealer sections, with subsections for four- and six-card hands. Four-card hands are included mainly to illustrate pegging tactics, while the six-card hands mainly deal with discarding problems and board strategy.

Like you, I try to analyze my games afterwards, particularly when I play against a computer program like Cribbage for Windows 97, which has a logging feature. If I find an improvement in my play, or a succinct demonstration of an important concept, I'll add a new record to my database.

Here are a couple sample entries from my database. The first is filed under pone six-card hands:

A-A-A-4-4-4

If you were dealt this hand, would you keep A-A-A-4 or A-4-4-4? Both are worth six points going in, and both will fetch twelve points on a ten-card cut. A-A-A-4 looks best since it has the lowest pip count, and generally lower cards are better. But it turns out A-4-4-4 is superior, since it gets twelve points on a 6, 7 or ten-card cut, while A-A-A-4 only gets twelve on a 9 or ten-card cut. A-A-4-4 is only worth considering for defense (A-4 is a tad safer toss to opponent's crib than A-A or 4-4).

Here's another entry, from the dealer four-card hands section:

3-3-4-J

I added this record as a result of a game I played at my Grass Roots club. The pegging started:

⁶  _J  ¹⁰  _?

I played my 4 here to bring the count closer to 31. But this left me holding an inflexible 3-3. A better idea would have been to play one of the 3s instead. This slightly increases the chances of giving up a go (it's vulnerable to an A or 2, instead of just an A), but it keeps the 5 trap alive. As it turned out, my opponent's last two cards were 5-6. If I'd had been holding 3-4 instead of 3-3, I'd have gotten a four-card run. Moral: as dealer it's worth taking small risks to save a 5 trap for the end of the play. Since pone will be naturally reluctant to toss you a 5, he'll often be holding one even if it doesn't match his other cards. As a result, a trap laid for a 5 has an unusually good chance of success.

Now I don't recommend you keep hundreds of records as I do. And it's not necessary to organize your database the same way I do. You could sort your entries by hand type, grouping together mid-card hands, or hands with 5s and ten-cards, or split lo-hi hands such as  A-4-x-x or 2-2-3-x. You can also organize your database by theme. For example, a section on 5 traps could include hands like 2-3-4-9, 4-5-6-6 and 6-7-8-x, in addition to the above 3-3-4-J example. And you could have other sections on magic elevens, different kinds of endgames, and so on. I do suggest, though, that you start with hands and plays that you find troublesome. As you continue your cribbage growth, add entries that communicate essential ideas in ways that are meaningful to you. The A-A-A-4-4-4 example I gave above helps me to remember that lower cards are not necessarily better, while the 3-3-4-J example reminds me to stay alert for 5 traps. You can rearrange your entries as your study focus changes, and delete records that become redundant or less useful. Soon you'll have developed your own personal theory of cribbage.

29 hands

Jack Remlin of South Hadley, Massachusetts asks:

"Can you tell me where to find accurate odds of getting a 29 point hand in cribbage? Are the odds the same for two-player and three-player games?"

Jack, the mathematical probability of getting a 29 hand in two-player cribbage has long been known to be 1 in 216,580. But I've never seen an accurate quoted figure for a 29 hand in three-player cribbage, so I decided to calculate it for myself. To understand the applicable math, I'll begin by demonstrating how the 1 in 216,580 figure for two-player cribbage is derived.

The branch of mathematics known as combinatorics provides a formula for calculating the number of possible combinations of k items selected from n total items:

In two-player cribbage, a hand contains six cards selected from 52 total cards, so according to this formula the total number of possible hands is:

Any six-card hand with the potential for 29 points must include a specific four-card combination: three 5s and the J of the missing suit. There are four such combinations available, (one for each J suit). The fifth and sixth cards can be anything else except the missing 5 (which must be the starter), so the number of eligible six-card hands is:

Now 20,358,520 divided by 4,324 is about 4,708.26, so your odds of drawing an eligible six-card hand are 1 in 4,708.26. Assuming you then keep 5-5-5-J, the odds of cutting the right 5 to it are 1 in 46. So your odds of getting a 29 hand in two-player cribbage are 1 in:

Or 1 in 216,580.

In three-player cribbage, you're dealt only five cards, so the total number of possible hands is:

Again, a potential 29 hand must contain one of four possible combinations of three 5s and the J of the missing suit. The fifth card can be anything else except the missing 5, so the number of eligible five-card hands is:

2,598,960 divided by 188 is 13,824.26, so your odds of drawing an eligible five-card hand are 1 in 13,824.26. Assuming you then keep 5-5-5-J, your chances of cutting the right 5 to it are 1 in 47 (not 1 in 46 as in the six-card game). So your odds of getting a 29 hand in three-player cribbage are 1 in:

Or 1 in 649,740. The odds are identical in four-player cribbage.

My thanks to Hal Mueller and Nick Wedd for their mathematical guidance.

Oops!

Bill Borghard wrote in with this:

"Terrific Web page! Awesome! I had started doing something similar to what Hessel had already done [i.e., calculate discard averages]. Kind of relieved, it would have been a lot of work. I did however, come up with a slightly different number of possible six-cards hands, ignoring suits. I have 18,395 instead of the 18,564 number. You can have:

4,1,1 (four of one rank, one of another and one of yet another)		=		=	858
4,2		=		=	156
3,1,1,1		=		=	2,860
3,2,1		=		=	1,716
3,3		=		=	78
2,1,1,1,1		=		=	6,435
2,2,1,1		=		=	4,290
2,2,2		=		=	286
1,1,1,1,1,1		=		=	1,716
Total:					18,395

Not much difference, but thought I would let you know."

The 18,564 figure was cited in the original posting of my article Discarding to your crib. You're right: the number of six-card hands irrespective of suit is actually 18,395. The error was mine, not Hessel's, and I've updated the article with the correct figure. Thanks for pointing this out!

Nice words

Linda Falkenstein writes:

"Thanks so much for your pages. It's so well done, exactly the kind of information I have been looking for. I look forward to your next update."

And Wilf Kelly, an occasional opponent of mine at PlaySite writes from Ontario, Canada:

"I enjoyed reading your articles on your site. I find it very interesting. I wish we had some tournaments in Canada, like you do in the States. Keep up the good work, and good luck next time we meet in crib."

Thanks for the kind words Linda and Wilf. As for sanctioned tournaments in Canada, that's starting to happen, at least out West. The ACC sanctions two yearly tournaments in Edmonton, and there was one last year in Prince George as well. There are also Grass Roots clubs in Alberta and British Columbia. Hopefully interest will spread eastwards. Perhaps it's time for the ACC to change its name to better reflect its international scope.

Mike Fetchel from East Hartford, Connecticut writes:

"Just wanted to drop you a line telling you that I really enjoyed your cribbage site. I've been playing cribbage for over 25 years and just recently joined the ACC and a Grass Roots club here in East Hartford. I'm always trying to improve my game and your site has been a big help. Thanks again and happy pegging."

Mike, I hope you enjoy playing competitive cribbage in the ACC. They're a tough bunch of players — the world's best. Good luck, and perhaps we'll meet at the Tournament of Champions in Reno one day!

John T. Irving, an old Internet adversary of mine from Vancouver, British Columbia, writes:

"Just finished reading your article A course of study and found it very informative. I plan on reading the books you suggest and look forward to your future articles on cribbage. Hope this will help get me to the next level. Look forward to playing again sometime."

John, you're the only person who ever beat me 6-0 in a Burbs match by skunking me twice in two games! I see you've pushed your Zone cribbage rating over 2600, so it looks like your game's stronger than ever. I'm glad the articles have helped your play, and look forward to our next match.

That's all for now. Keep in touch everyone, and we'll do another one of these soon. 

- January 2001