Schell's mailbag, January 2001

This month marks the first anniversary of
Cribbage Forum, which went online in January 2000. In observance of this
milestone, I'm presenting the first installment of Schell's mailbag,
devoted to your questions and comments.RulesDavid Steinbrenner from Wynnewood, Pennsylvania writes:
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:
If you play a card that makes the count 31, what you get is:
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! DiscardingKlagett, from Upstate New York, submitted this via the Yahoo! Cribbage Message Board:
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, 245810J 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 24 or 28 from this hand. Let's see how these two discards perform relative to each other, and relative to the other alternatives, including your own 10J toss:
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 24 and 28 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 24 returns a higher expected average than tossing 28, 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 28 will get you about ¼ point less between hand and crib, but it leaves you with 4510J, which is a slightly better offensive pegging hand due to its two sets of touching cards. 4510J pegs one point more than 5810J if pone has four tencards. And you might even get lucky and trap pone's 6 for a 156 at the end of the play. If you're playing off however, definitely keep 5810J. The 8 gives you a comfortable reply to a low card lead, something that's hard to defend against holding 4510J. Note that if you apply the addition method to this hand, you'll be correctly guided to the 5810J 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 25810, keeping four wellspaced 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 2458, 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:
Hey Ras, thanks for the kind words, and congratulations on your fast start to the 20002001 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. PeggingSeveral readers wrote in with pegging vignettes. A Yahoo! Games player named niner1 writes:
Hey, congratulations on your fast start at Yahoo! Regarding your question: as you've apparently noticed, the problem with leading a 2 from 289J 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 152 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 tencard 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 threeonone 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 114109*, 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 89 for last, hoping to trap dealer into a run later. Now imagine the same hand and cut, but at 116118*. 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 twopoint 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:
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 AAA8. 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 4567. 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 twopoint 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:
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:
Personal databaseDuane Stansbie from Toowoomba, Australia asks:
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 sixcard hands. Fourcard hands are included mainly to illustrate pegging tactics, while the sixcard 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 sixcard hands:
If you were dealt this hand, would you keep AAA4 or A444? Both are worth six points going in, and both will fetch twelve points on a tencard cut. AAA4 looks best since it has the lowest pip count, and generally lower cards are better. But it turns out A444 is superior, since it gets twelve points on a 6, 7 or tencard cut, while AAA4 only gets twelve on a 9 or tencard cut. AA44 is only worth considering for defense (A4 is a tad safer toss to opponent's crib than AA or 44). Here's another entry, from the dealer fourcard hands section:
I added this record as a result of a game I played at my Grass Roots club. The pegging started:
I played my 4 here to bring the count closer to 31. But this left me holding an inflexible 33. 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 56. If I'd had been holding 34 instead of 33, I'd have gotten a fourcard 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 midcard hands, or hands with 5s and tencards, or split lohi hands such as A4xx or 223x. You can also organize your database by theme. For example, a section on 5 traps could include hands like 2349, 4566 and 678x, in addition to the above 334J 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 AAA444 example I gave above helps me to remember that lower cards are not necessarily better, while the 334J 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 handsJack Remlin of South Hadley, Massachusetts asks:
Jack, the mathematical probability of getting a 29 hand in twoplayer 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 threeplayer 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 twoplayer 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 twoplayer cribbage, a hand contains six cards selected from 52 total cards, so according to this formula the total number of possible hands is:
Any sixcard hand with the potential for 29 points must include a specific fourcard 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 sixcard hands is:
Now 20,358,520 divided by 4,324 is about 4,708.26, so your odds of drawing an eligible sixcard hand are 1 in 4,708.26. Assuming you then keep 555J, the odds of cutting the right 5 to it are 1 in 46. So your odds of getting a 29 hand in twoplayer cribbage are 1 in:
Or 1 in 216,580. In threeplayer 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 fivecard hands is:
2,598,960 divided by 188 is 13,824.26, so your odds of drawing an eligible fivecard hand are 1 in 13,824.26. Assuming you then keep 555J, your chances of cutting the right 5 to it are 1 in 47 (not 1 in 46 as in the sixcard game). So your odds of getting a 29 hand in threeplayer cribbage are 1 in:
Or 1 in 649,740. The odds are identical in fourplayer cribbage. My thanks to Hal Mueller and Nick Wedd for their mathematical guidance. Oops!Bill Borghard wrote in with this:
The 18,564 figure was cited in the original posting of my article Discarding to your crib. You're right: the number of sixcard 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 wordsLinda Falkenstein writes:
And Wilf Kelly, an occasional opponent of mine at PlaySite writes from Ontario, Canada:
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:
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:
John, you're the only person who ever beat me 60 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 
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