Eolus619 says: you ask a good question. Liam Brown tables worth an educational “look see“ as to answers when considering hand + crib combined values ...have to see what Hal says about pegging

Rosemarie44 says: I see that this hand is more defensive not realizing this was the more deciding factor in retaining these cards.

Eolus619 says: your explanation is food for thought ..at least for me !..thanks

Eolus619 says: morning ...my question to you is about 5s. Specifically as dealer , is your discard decision about where the 5 goes...based on each individual hand makeup combined with board position ..or do you have a more philosophical decision guideline about 5s that you make most of the time like Ras about flushes ..when in doubt FLUSH ..which morphs to “the five almost always seems to do better in the dealer crib “ so send it there ..thanks

Eolus619 says: morning...i do not understand “account for nine pegs”..could you please expand on this . thanks

Ras2829 says: Hi cwed: To own crib, the 5-8 average is 5.564 (550) 16/91 and A-4 average is 5.416 (1,771) 22/91. Both discards score 8 or more points 26% of the time. The big difference is A-4 will score a mere two points 80% more frequently than will 5-8 based on my empirical data. Note that I've tossed the A-4 1,771 times and recorded the results. What about the 8-10 here? More later!

Eolus619 says: hello....you will most definitely benefit from comments of a more experienced player than me. Having now read the comments above and looked at Ras’s dealer discard table it seems the 5-9 averages .13 less than 5-8. But the new combination and an A-4 discard is rated most total points on Brown with 5-Q and 5-9 close. But still the only apparent reason to not discard a 5 combination would be Dan’s explanation. However, please remember i am a player who buys his return bus ticket before he pays his tournament entry fee knowing the likely outcome of my play...all the best

JQT says: This posted solution beautifully shows the power of Halscrib, revealing that after Toss (5 8), we shall expect to enjoy a difference of approximately 0.28 points of relative gain on the Cribbage Board. Some of this will come from what may initially appear to be our detriment, which involve a Hand that upon average is 0.09 points weaker, but then it also includes a gain from enduring an average of about 0.11 fewer points lost during the pegging (a defensive advantage), as well as an average 0.26 points more from the Crib itself. Thus, in aggregate, taken in reverse order respectively, 0.26 plus 0.11 minus 0.09 equals an overall advantage of about 0.28 points, or greater than a quarter-of-a-point, after Toss (5 8), as opposed to Toss (A 4). Of course, these numbers would all be calculated to be slightly different if a DEFENSIVE strategy had not been determined as best here by the program, since it also can converge upon either an OFFENSIVE or an OPTIMAL or a SAFE strategy as well, depending on the Relative Board POSITION. This program plays an extremely powerful game, and while it can be bested by a few very proficient players in a series of alternate-first-dealer of say 500 or 1000 games, it is definitely among the strongest opponents to have ever played 'This Game of Ours' fondly known as Cribbage. If and when the computer program and its inherent algorithms does make a 'mistake' or lead to an inferior conclusion, it would least likely be from its Hand Evaluation, which, since all of the data used to generate this are 'known' when this is tallied, it is exactly the same number for any program, and is essentially the summation of all possible Hands from all possible Cuts after being divided by the number of possible Cuts (46 Cuts, or 52-card deck, minus the 6 Cards dealt and thus known to us during each deal). The first area in which any program (and including of course us human players) might make a small miscalculation would be from the Crib Value, a value that is often taken from either computer-generated tables, or from empirical data gathered by machine or human players. The tables used by a strong program such as Halscrib are often part of 'secret sauce' in a program, and have and will evolve over time, as more accurate data can be gathered. We often refer to Schell's Tables as being quite good values, and these are actually a composite of data from other tables, the most notable being those tabulated by Hessel, Colvert, and Rasmussen (empirical). It is my understanding that Halscrib uses tables that are an amalgamation of several of tables such as those just mentioned, along with some additional 'tweaks' carefully decided by the program's author. But rest assured that any differences among these tables or any that might be refined and improved upon in the future shall only involve second- and most likely third-decimal place differences, and so this, which is all very interesting and fascinating, is an unlikely place to look for any vast enhancement and thus improvement in future machine or human play. Next, we might examine the numbers that are derived from Pegging. It should be noted that many sites that evaluate Cribbage Hands not only ignore POSITION on the board, but also ignore the PEGGING! This is because it is no trivial matter to evaluate the intricacies of pegging in Cribbage. Notice that in many relative positions, especially those that are DEFENSIVE in nature, often involve not us pegging more, but actually the RELATIVE AMOUNT between what is pegged by our Opponent and Us being less! Incidentally, this is the one area in which expert players look most closely to improve their game: it involves not just the nuances of pegging, which are enormously intricate, but also in assessing the nature of Relative Position, even as today when we are at the Initial Position of (0*-0)! Lastly, I believe that the most suspect errors and blunders come from simple human error itself: We get tired, and forget whose Crib it is, and place the wrong cards into the Crib! Or we hold an 8 Card and in our tired mind, and blurred vision, we might see just half of it and think it's a 3 Card! And such human error could also include looking at the wrong column of data in a Crib Value table, or showing the wrong data from a program's data. And this leads me to one final point: I should note that after seeing yesterday's Halscrib Analysis, I do believe the data was either not input correctly to the program or the solution was not enumerated correctly, and I'll reveal why. After performing several playouts with a real deck and real board, the hand displayed and chosen as best fails to win at a dismal rate! I should stress that I could very well be wrong, but having created this puzzle, I do feel a certain responsibility to both the composition of it, along with what I believe to be its proper solution. And since I doubt that the program is revealing an error of a 'bug' that this instead could more typically involve something well-understood by the programming field as "The Horizon Effect" and thus would involve some minuscule miscalculation, I suspect that maybe the wrong column of data was chosen to be used as the one that will properly decide the outcome of the game (I am quite confident that our ability to DEFEND and then go out via First Hand Show is far more relevant than any other factor, and so retaining any card close in rank to our PAIRS Royal allows the Dealer to quite easily grab a RUN and thus obtain Three Holes before we could ever hope to peg TWICE that much). It's hard for me to believe that "The Horizon Effect" could ever misjudge an endgame by such an astounding amount, but I have seen the equivalent occur in Chess Programs. Let me be clear however: very likely, I am just wrong, and even with my extensive care in both composing and testing this puzzle, human error on my part is the most logical place to look for any reasons for such a discrepancy. But the data could have also been either entered or tabulated incorrectly, or less likely, it could actually be an illustration of "The Horizon Effect." I hope this little diatribe of mine is not confusing or boring, but having spent much of my life enjoying this wonderful game, and having composed the puzzle, I am concerned that something just doesn't seem to 'add up' in the computer program's solution. I know that many people who are new to this site often view the advice and the solutions of either certain human players, or of the computer program, to be DEFINITIVE of being correct, and yet as much as we often agree, there are times when all of these will disagree. It's what makes Cribbage such a wonderful game, I believe. Even during years of puzzles being examined on Schell's wonderful "Ask The Experts" pages on CribbageForum dot Com, some of the top players in this game, along with the computer program, often disagreed. I firmly believe that this means we should not take *any* solution as being 100% best, but should always keep an open mind. As I said earlier, it should probably not be surprising that most of the errors I have encountered and discovered in Cribbage, as well as in Life, have one thing in common: more often than not, the errors come from and belong to me! - j q t -

JQT says: Good Evening, RAS! I enjoyed your puzzle, as always, and I apologize that my little dissertation (above) has 'Pushed your Posting' to the Ends of the Earth! And so, maybe it's a convenient time to 'pile on even further' and ask a late-evening pegging question! Something I often find curious during the pegging with such hands as we've kept today is not just when to 'unload' the Lone Ace, which can be a pegging liability, but also, when and which Ten Card or "X" to rid ourselves of FIRST. For example, after a 6 Card Lead by Pone, we may first notice and then be a bit concerned at this 'show of strength' alongside the 7 Card Cut; but now, do we 'unload' the possibly-burdensome Lone Ace, or do we play a Ten Card, and if so, which one do we play FIRST? This latter theme is what I am most curious about, since frequently I see many players respond with what is agreed upon to be the 'safer' "X" Card first, which is the Queen. And yet I would instead almost always 'dump' the Ten Card "T" first, as I think it can actually be more easily 'trapped' into a RUN for our Opponent during a later volley of the pegging. What, if any advice or experience, would you share with regards to the order in which we rid ourselves of such Ten Cards (or "X") during a Defensive Pegging scenario such as this? And as an added Bonus Question, would you be even more concerned today with ridding your Hand of the "Lone Ace," before concerning yourself with the order in which to dispense of those Ten Cards, "T" and "Q"? Thanks again for posting such excellent puzzles and thorough ideas, and for also taking such an active role in being a Moderator for the site, which helps in 'feeding us' such a good stream of 'healthy, as well as informative, and also instructional' puzzles, day after day. This does not go unnoticed!

Ras2829 says: Hi JQT: And thanks for the kind words. This game and those who play it have been such a wonderful part of my life, there is no way that I can fully repay the value in a single life span. Thanks to a forgiving Lord and the advances of heart medical science, all years beyond 33 have been a gift.