Monday 3:10 AM

Cutting Jacks and the Deep Dark Nines

We need a Dozen Points to WIN this Cribbage Endgame Puzzle by UnclePegg, and that's a Special Case Scenario, so let's map out all of the possible ideas that might see us cross the Finish Line. Yes, this means we'll look at EVERY POSSIBLE DISCARD! (And try to keep it under three thousand words, for Goodness Sakes!) ☕ 🍩

This discarding analysis endeavor would normally be an arduous annoyance and tedious task, since '6 choose 2' equals fifteen unique discard decisions, but having been dealt Three 9 Cards and Two 5 Cards, we can and will ignore suits, and thus the complex combinatorial conundrum reduces itself down rather nicely to become a significantly less unwieldly 'brouhaha' of just FIVE choices!

With the Dealer needing only Thirteen Points (Three Points BELOW Dealer Average) to defeat us, we won't initially concern ourselves too much about what we put in the Crib, so we'll temporarily ignore the respective Crib Value and Total NET score for each idea, because if we cannot go out with First Hand Show, it's very likely that we'll LOSE this game anyway, regardless of what we place in the Enemy Crib.

However, in order to be thorough and comprehensive, we will of course look at the remote possibility of what happens if the Dealer comes up "short," and thus allows us to possibly peg out as the Next Dealer, when we would actually have the Pegging Advantage. Note that both a Jack Cut and a 5 Card Cut can often drastically affect an Endgame Pegging Battle in Cribbage, so try to keep these in mind today, and I'll comment further down (*) about this.

We'll proceed to look not only for the ability to score those Dozen Points via the Cut Card, but we'll also project our chances of winning if we can attain the Pone Pegging Average of about Two Holes, and therefore we'll look both actively and precisely for ways that we can obtain Ten Points or more from the Cut Card. Okay, on with the Hand Mapping!

* * *

(109-108*) (5h 5d 9s 9c 9d Tc) Cut = 6c, by UnclePegg, Oct 7, 2024.

Dump a PAIR of 9 Cards Idea:

Keep (5h 5d 9d Tc) and Toss (9s 9c)

Expected Hand Average is 408 DIV 46 = 8.870
(minus) Expected Crib Value (Schell) is 6.390
Total is 2.480

02x14=028 - 55 (Maximum Hand, 2 cuts = 14 Points)
04x13=052 - JJJJ (6 cuts >= 13 Points)
03x12=036 - TTT (9 cuts >= 12 Points)
12x10=120 - AAAA, QQQQ, KKKK (21 cuts >= 10 Points)
04x09=036 - 8888 (25 cuts >= 9 Points)
05x08=040 - 6666, 9 (30 cuts >= 8 Points)
16x06=096 - 2222, 3333, 4444, 7777 (Minimum Hand, 16 Cuts = 6 Points)
46 ___ 408

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Dump the PAIR of 5 Cards Idea:

Keep (9s 9c 9d Tc) and Toss (5h 5d)

Expected Hand Average is 388 DIV 46 = 8.435
(minus) Expected Crib Value (Schell) is 9.370
Total is -0.935

08x15=120 - 8888, JJJJ (Maximum Hand, 8 cuts = 15 Points)
05x12=060 - 6666, 9 (13 cuts >= 12 Points)
05x08=040 - 55, TTT (18 cuts >= 8 Points)
28x06=168 - AAAA, 2222, 3333, 4444, 7777, QQQQ, KKKK (Minimum Hand, 28 Cuts = 6 Points)
46 ___ 388

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Toss (5 T) Idea:

Keep (5d 9s 9c 9d) and Toss (5h Tc)

Expected Hand Average is 364 DIV 46 = 7.913
(minus) Expected Crib Value (Schell) is 7.41
Total is 0.503

09x12=108 - AAAA, 6666, 9 (Maximum Hand, 9 cuts = 12 Points)
17x08=136 - 55, TTT, JJJJ, QQQQ, KKKK (26 cuts >= 8 Points)
20x06=120 - 2222, 3333, 4444, 7777, 8888 (Minimum Hand, 20 Cuts = 6 Points)
46 ___ 364

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Toss (5 9) Idea:

Keep (5d 9s 9d Tc) and Toss (5h 9c)

Expected Hand Average is 312 DIV 46 = 6.783
(minus) Expected Crib Value (Schell) is 6.150
Total is 0.633

04x12=048 - JJJJ (Maximum Hand, 4 cuts = 12 Points)
04x10=040 - 8888 (8 cuts >= 10 Points)
14x08=112 - AAAA, 55, 6666, 9, TTT (22 cuts >= 8 Points)
08x06=048 - QQQQ, KKKK (30 cuts >= 6 Points)
16x04=064 - 2222, 3333, 4444, 7777 (Minimum Hand, 16 Cuts = 4 Points)
46 ___ 312

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Toss (9 T) Idea:

Keep (5h 5d 9d 9c) and Toss (9s Tc)

Expected Hand Average is 308 DIV 46 = 6.706
(minus) Expected Crib Value (Schell) is 5.430
Total is 1.266

04x12=048 - AAAA (Maximum Hand, 4 cuts = 12 Points)
02x10=020 - 55 (6 cuts >= 10 Points)
20x08=160 - 6666, 9, TTT, JJJJ, QQQQ, KKKK (26 cuts >= 8 Points)
20x04=080 - 2222, 3333, 4444, 7777, 8888 (Minimum Hand, 20 Cuts = 4 Points)
46 ___ 308

* * * 🌼 * * *

Let's look at the results of our Hand Mapping, and examine some of the nuances of what we have found:

Toss (9 9) has Nine Cuts (55, TTT, JJJJ) that yield a Dozen Points or more, and this means that 9/46 equals 0.196 or close to about 20% of the time, we won't have to peg a thing!

We also have another Dozen Cuts (AAAA, QQQQ, KKKK) that give us Ten Points, so if we combine this with the previous calculation, this means that after Toss (9 9), if we can peg the average of Two Holes, we should be able to WIN about 21/46 equals 0.457 or greater than 45% of the time.

* * * 🌼 * * *

Toss (5 5) it seems as though we shall ostensibly test this idea just for laughs, and yet surprisingly it has Thirteen Cuts (6666, 8888, 9, JJJJ) that yield a Dozen Points or more, and this means that 13/46 equals 0.283 or close to about 28% of the time, we won't have to peg a thing!

Sadly, it has no Cut Cards that yield Ten Points, as it "gaps" all the way down to Eight Points for the next helpful cut, and since it's very unlikely that we'll be able to peg Four Holes, we won't be "stooping" this low. But who would have thought that Toss (5 5) had a 40% better chance for simply producing an "Instant Winner" of a Dozen Points or more?!

* * * 🌼 * * *

Toss (5 T) has just Nine Cuts (AAAA, 6666, 9) that yield a Dozen Points, and this means that 9/46 equals 0.196 or nearly 20% of the time is how often we'll score a Dozen.

Sadly, like Toss (5 5), if we did Toss (5 T), there are no Cut Cards that yield Ten Points, as it "gaps" all the way down to Eight Points for the next helpful cut, and since it has fewer Dozen-Point "Instant Winners" than Toss (5 5), we won't have to examine Toss (5 T) any further.

* * * 🌼 * * *

Toss (5 9) has just Four Cuts (JJJJ) that yield a Dozen Points or more, and this means that 4/46 equals 0.087 or barely 9% of the time is as much as we'll score a Dozen.

And in addition, we have another Four Cuts (8888) that give us Ten Points, so once we combine this with the previous calculation, this means that after Toss (5 9), if we can peg the average of Two Holes, we'll only be able to WIN about 8/46 equals 0.174 or a scant 17% of the time.

Toss (5 9) is a poor choice not only because of its low-scoring potential, but also because it only begins with Four Points. This means that even if we defended well and brought this game to another deal, after Sixteen Cuts (2222, 3333, 4444, 7777) or a massive 35% of the time, we probably won't get close enough by completion of this deal to peg out as the Next Dealer. Thus, we can confidently "discard" this Discard Idea!

* * * 🌼 * * *

Toss (9 T) also has just Four Cuts (AAAA) that yield a Dozen Points or more, and this once again means that 4/46 equals 0.087 or barely 9% of the time is as much as we'll score a Dozen.

And in addition, we only have another Two Cuts (55) that give us Ten Points, so once we combine this with the previous calculation, this means that after Toss (9 T), if we can peg the average of Two Holes, we'll only be able to WIN about 6/46 equals 0.130 or a paltry 13% of the time.

Toss (9 T) is therefore a poor choice not only because of its low-scoring potential, but also, because like Toss (5 9), it only begins with Four Points. This means that even if we defended well and brought this game to another deal, after a colossal Twenty Cuts (2222, 3333, 4444, 7777, 8888) or greater than an immense 43% of the time, we likely won't get close enough by completion of this deal to peg out as the Next Dealer. Thus, we can quite readily also "discard" this Discard Idea, and place it on the bottom of the heap of poor candidates!

* * * 🌼 * * *

After combing through all of the data very carefully, let's make an informed, well-researched Discard Decision and Toss (9 9) today, and just like some Rough-Cutting Rap Lyrics might tell us, "Pay the Dues, Forget the Fines; Rid Yourself of the Deep Dark Nines." 🌟 If we restate our findings from above in less lyrical form, we may recapitulate:

Toss (9 9) has Nine Cuts (55, TTT, JJJJ) that yield a Dozen Points or more, and this means that 9/46 equals 0.196 or close to about 20% of the time, we won't have to peg a thing!

We also have another Dozen Cuts (AAAA, QQQQ, KKKK) that give us Ten Points, so if we combine this with the previous calculation, this means that after Toss (9 9), if we can peg the average of Two Holes, we should be able to WIN about 21/46 equals 0.457 or greater than 45% of the time.

The astonishing finding is of course that Toss (5 5) has an even-higher chance of producing an "Instant Winner," as it has Thirteen Cuts (6666, 8888, 9, JJJJ) that yield a Dozen Points or more, and this means that 13/46 equals 0.283 or close to about 28% of the time, we wouldn't have to peg a thing!

But sadly, as we already noted, it has no Cut Cards that yield Ten Points, as it "gaps" all the way down to Eight Points for the next helpful cut, and since it's very unlikely that we'll be able to peg Four Holes, but it is likely that we'll peg at least something, we should be evaluating this as more than a discrete WIN/LOSE proposition. We have a slight amount of "Wiggle Room," since we might peg a bit, and the Dealer might also come up "short."

Those additional Dozen Cuts (AAAA, QQQQ, KKKK) that gives us Ten Points after Toss (9 9) provide a huge boost for our chances of winning, and in addition, Toss (9 9) means we're very likely generating LESS HELP in the Enemy Crib, so it will also increase the odds that we might survive until another deal, if one is necessary.

But once again, who would have thought that Toss (5 5) had a 40% better chance for simply producing an "Instant Winner" of a Dozen Points or more today?! If you have zero confidence in your pegging skills at this position and with these cards, Toss (5 5) is NOT an unreasonable Discard Choice today! 🥰

It's interesting to note right from the start that if we get NO HELP after Toss (9 9), if we can peg PAIRS Royal for Six Points, our Hand of Six Points will WIN the game, even with NO HELP whatsoever from the Cut Card. Having a PAIR of 5 Cards gives us a chance to do this, so we should not be discouraged, even if we obtain a LOSER Cut Card today.

After the 6 Card Cut, which is not quite LOSER status, we now have Eight Points in our Hand, and we either need to peg Four Holes during this deal, or we need to survive the onslaught of the Dealer's combined Pegging, Hand, and Crib, and hope that the aforementioned mess 'adds up' to be fewer than Thirteen Points!

It's actually not an easy calculation of whether we should try to peg aggressively or defend, because we really want to do BOTH. I categorize such instances of Cribbage Endgame Strategies as "Opportunistic Pegging," which means that if the Dealer allows us a Run or PAIR, we should probably take it!

However, we should also keep in mind that since we need to peg Four Holes to WIN during this deal, it may be to our advantage to peg NOTHING AT ALL as opposed to pegging Two Holes, if in doing so, those Two Holes we get while pegging 'give up' the same number of holes or more to the Dealer.

Our chances of surviving to play yet another deal, while not great, probably exceed our odds of being able to peg Four Holes. It's crucial to understand this predicament, and all of the ramifications of giving up needless pegs to the Dealer at such a Relative Score, and with this Hand.

Why do I say "probably"? It's because that 6 Card Cut just "blew a gasket" in our hypothetically "Defensive" Discard of Toss (9 9), since (unbeknown to the Dealer, but KNOWN BY US) the Dealer's Crib now already contains at least nearly HALF of what the Dealer needs to reach the end of the board and defeat us if we cannot go out via First Hand Show!

(*) While we're "digging deep" today, give yourself Extra Credit if you noticed that after a Jack Cut, which is usually something very scary for us as Pone, since it will suddenly and immediately place the Dealer ahead of us and at Hole 110, that after Toss (9 9), and after Toss (5 5), and even after Toss (5 9), as Pone, a Jack Cut ALWAYS PUTS US OUT! It's almost comical, because if a Jack Cut does occur, both players might quietly be thinking, "My Opponent doesn't know it yet, but after that Jack Cut, I've now got enough to go out!" 😫

The Top Two Discard Ideas thus contain a Lethal Dose of "Jack Cut Insurance" today; that is, lethal for the Dealer! This is a very rare occurrence in Cribbage Endgame Battles, and credit goes to the puzzle composer, UnclePegg, for this special effect! We can only hope to see more interesting puzzles from this new contributor.

Let's look at the Three Options we have to start the pegging:

Leading a 5 Card from our PAIR would allow us to score after Ten Replies (AAAA, 55, 9, TTT) by the Dealer, and while it's very unlikely a Dealer would PAIR such a lead, this would allow us to WIN, unless of course (gasp!) the Dealer had the "Case" 5 Card!

But the 5 Card PAIR could be our Goal-Getter today, especially during the latter stages of the pegging, and 'Leading from Strength' is almost always a bad idea when the Dealer is DEFENDING. We want our FIRST 5 Card to appear in the Dealer's face when the options to duck and dodge it are limited, and this will usually NOT be the case when the Dealer still holds Four Cards.

Leading a 9 Card, often considered to be the Least Likely Lead Card in Cribbage, would allow us to score after Seventeen Replies (AAAA, 55, 8888, TTT, JJJJ) by the Dealer, and about Six (AAAA, 55) of these are likely to be played.

Leading a Ten "T" Card would allow us to score after Eleven Replies (55, 8888, 9, JJJJ) by the Dealer, and only Two (55) of these are somewhat-remotely-likely to be played.

With all that (and more) in mind, let's lead our 9 Card, and peg OPPORTUNISTICALLY! The 9 Card Lead does a few things, as it not only breaks up the 9-T duo, but it also has the most Dealer Responses that allow us to score.

Most importantly, we "Keep our Powder Dry" as our PAIR of 5 Cards is quite capable of bringing us a VICTORY during the pegging. Lastly, the 9 Card Lead is both DEFENSIVE and yet, after a 6 Card Cut, a show of strength. Since we sent a PAIR of 9 Cards into the Dealer's Crib, our impetus to peg out is now greatly elevated, because we've already contributed nearly HALF of what the Dealer needs in the Crib to "cover the distance" via our discard.

If the Dealer replies to our 9 Card Lead with a Queen or King, follow up with the Ten "T" Card Reply and 'Push the Count' up to Twenty-Nine. Now, if the Dealer has to lead from two or three cards, and ONE of them is a 5 Card, the Dealer may be forced or feel compelled to PAIR our 5 Card! And if we can peg PAIRS Royal today, the GAME IS OURS, as long as the Dealer cannot peg Thirteen Holes. 🍇 🍂

Wordle 1,206 3/6 (surprisingly still often sold by volume and not weight)

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Monday 3:12 AM

Great puzzle. I score a .02 difference tossing a pair of 9s vs. the 5s to the crib, the 9s being slightly higher. But, of course, for this outcome, the difference is the show card which pretty much guarantees a win for the pone if the 5s went to the crib. With the 9s toss, recovery requires opening with the 10.

Monday 4:51 AM

Monday 5:20 AM

Monday 5:28 AM

I've kept one of each suit.

All Ten #749
10/10
1✅ 2✅ 3✅ 4✅ 5✅ 6✅ 7✅ 8✅ 9✅ 10✅
https://beastacademy.com/all-ten

Monday 5:38 AM

Choosing offense seems the only option with dealer needing 13 to win. Unfortunately, these cards represent little in the way of pegging potential. On average will peg barely a single point. Leading the 9 offers best hope for win though not likely. N/D is likely to end game in hole 118 or 119. Take any pegs offered along the way. It is difficult for n/d to peg under best of circumstances; dealer will avoid pegs most of the time in this situation. Have to hope that dealer has a low scoring hand and a low-scoring discard to crib. Under such conditions dealer just might decide to peg somewhat aggressively. Even that is questionable as n/d needs to score 12 to win. What are chances that n/d can do that? That's one in four attempts. Outlook is increasingly bleak after the 6 cut as n/d knows of six points in crib. Not a good beginning to this hand and favors those who do not recall what they discarded. They'll not be stressed by those six points in crib. Ignorance is bliss.

Monday 5:38 AM

Thx for the puzzle UnclePegg..I am tossing two 9s..I am looking for a favorable cut from the 15 remaining Xs….the most frequent dealer discard is 7/8..so 🤞

Monday 5:50 AM

6-8-9-10-J vs A-5-8-X the second set is better. Relative positions vs pegging with the 5s . The 5s get the vote. Nine lead hope to use the fives to peg four. dec

Monday 6:44 AM

I like it this way best. But I should have kept the 9s. Oh well.

Monday 7:38 AM

If I knew the 6 would be the starter ... Ugh.

Monday 9:23 AM

Not holding 9s--about as bad as fce cards for end game pegging. Hoping for one of the faces and pegging 2--or on if I get the right jack lol

Monday 10:32 AM

This was really my only choice in this potion. I will the 10.

Monday 10:32 AM

5-5 for me today. Yesterday I did indeed accidentally discard as if I was the dealer instead of as the non-dealer. Thank you to those that provided me with creative criticism / comments.

Monday 11:32 AM

At this point in the game, have to keep the 5-5-10 intact, as it’s our best hope of getting a cut and going out. If only I had X-Ray vision to see the cut card…kind of like in Austin Powers!

Monday 11:45 AM

Chances are Dealer is going out anyway if I don't so I'm not going to worry about what I throw to opponent's crib. I'll count the cuts 10pts+ for 5-9-9-10, 5-5-9-10 and 9-9-9-10:

5-9-9-10

10pts - 8888 = 4 cuts
12pts - JJJJ = 4 cuts

5-5-9-10

10pts - AAAA, QQQQ, KKKK = 12 cuts.
12pts+ - 55, 101010, JJJJ = 9 cuts

9-9-9-10

12pts+ - 6666, 8888, 9, JJJJ = 13 cuts.

Although 9-9-9-10 has 4 more outright winning cuts than 5-5-9-10, the latter has more cuts with a chance of going out with the pegging and the 9-9 throw gives us more of a chance of it going to another deal.

So I'll throw a 9-9.

Monday 11:45 AM

At 109-108* playing an Offense strategy for the pegging the Win/Loss %s are:

Offense_____________Win %_______Loss %
5-5-9-10_____________40.1________59.9
9-9-9-10_____________30.0________70.0
5-9-9-9______________21.7________78.3
5-9-9-10_____________21.6________73.5

5-5-9-10 is very much best for Win %s and lowest for Loss %s so I'll select 9-9 to discard.

After the 6 cut I'll lead the 9 and play Offense:

Lead______________Our Pegging Points
9________________________1.29
10_______________________1.25
5________________________0.98

Monday 6:05 PM

Tough puzzle today! :X

Monday 7:33 PM