Friday 3:15 AM

Our deal, our crib. I was leaning towards the Flush. Twenty-eight cuts keeps the hand at 12 points (2 at least in crib).. More cuts improve the value of this hand. Still not 100%.

   

Friday 3:36 AM

Of course if that cut gives our opponent a twelve to fourteen point hand we might be sunk. I went for 16-23-24 points hand. They lead 6-7-8 We might just be in a lil trouble here. dec

   

Friday 3:51 AM

This problem is solvable.

That is to say there is exactly one correct hold and all others are wrong. Following this hold it might be that the pegging is subject to some degree of randomness so as not to be predictable, but this is also solvable.

Unfortunately we will not find out today which is the right answer and which is wrong.

There is a discussion of this here:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5826371/

In terms of the poker bot Cepheus, which is available online and will give the correct play for given hands. http://poker.srv.ualberta.ca/

It does this by considering all possible bets and cards for two handed limit poker.

Cribbage turns out to have a high number of decision points.

Those we can see today are the score, starting hands, cut and then pegging options.

There are 121^2 scores possible plus two 'game over states'.

For an individual deal the naive calculation of the complexity of cribbage is

52C6 * 46C6 * 40C1, i.e. 7.63 * 10^15 starting hands.

There are then 6C4 = 15 discards per player

Today we can consider the 4567 straight flush, 3456 straight flush , the 4556 double run, among others.

Therefore solving this puzzle today we would consider the outcomes for each possible six card pone hand, and then perform a learning process for each discard to find out the outcomes.

The problem making cribbage more complex than poker is that the 121^2 score positions are all different .

Today at 91*-105 for example it's clearly a different puzzle to 91*-111. Pone needs 16 points and probably won't get then here but is almost certain to get them next time if we don't go out.

We need 30 points.

Each cribbage starting position is logically a unique position. For example, we can train a cribbage bot that makes the most net mean points. This is effective for poker since the bot simply needs to win money, and it's easy to optimise for that goal.

For cribbage this won't be correct.

If we consider the 15 discards and 46C6 opponent hands and the cut, then pegging is then a game with 4!*4! = 576 possible plays. That is to say that there are 4 choices for the first card, three for the second and so on.

Even though pegging doesn't always follow turn order after the first three cards, the players cannot specify turn order, so there are 'only' 576 pegging routes for two hands. In many cases less, as if we hold TTTA and dealer holds JJJ9 then the combination TJTT is impossible.

However it's probably sufficient to consider this as 576 decision points.

Having done so we come to the problem that for example at 0-0* there's no reason to be certain that a discard that averages net +4 is better than one that averages net +3.5, since the +4 average could hide some bad cribs.

Optimizing for points won't be BAD but it will not be 'solved'.

Solving cribbage is hard because while winning is a goal, and we can resolve backwards each point pair from 120-120 for all possible starting hands, there are other goals such as winning a skunk (which we might value at $2 Vs $1 or 3 points Vs 2), or double skunk. In some cases we might ONLY want a skunk and this implies a different strategy from 0-0.

Anyway today we have an average of around 19.5 points and then 3.5 maybe from pegging (note that pone is unlikely to want pegs unless he has a big hand, which probably he does not, so he will peg defensively).

This would put as around 114 points next time, which is well within striking distance to win, which is generally 8-10 points. Of course even at 114 we often lose.

Here my feeling given these numbers is that it might be best to go for the balanced points

Which today is the 3456 straight flush! The flush starts at 12 and averages 13.9.

That says 'this hand doesn't improve much'

Meanwhile the 4556 double run starts at 12 Me averages 16, which might be from some 24 point hands.

But I'm not sure those hands help so much as if we score 12 from our hand and then maybe zero in the crib then we are stuck on 106 or so, next time and praying to the cribbage gods for a magic hand.

So cancel my vote and go for the 3456 . With 57 to the crib we get at least some points there.

But we won't know for sure.

Anyway... As per the above the mathematical complexity of cribbage is high. I think perhaps I could solve for the -120* case, in which case suits are irrelevant as the hands never get laid down.

In answer to my quiz question yesterday, the furthest out I think you could win from seems to be

5-3-4-a-2-7-6 scoring 3, 7, 7 = 17 points

Obviously that score is in fact solved... We can say that there are no deals where dealer can lose from 104-120*, since the first run of three is defeated by playing the A , which makes it impossible for pone to score the full 17 points.

The follow-up question would then be 'what is the furthest away that pone can win from at -120* with perfect discarding and obstructive play by dealer?' for example let's say we are dealt 444555 as dealer at -120*. We can only hold 4555, 5554, or 4455.

Clearly we may be powerless to lose at some x-120* scores , even if it takes a 1 in a million deal! But I'm not sure what scores they are....

   

Friday 4:05 AM

Interesting puzzle today-always tough to break up the 4556 as deaker,even with a couple appetizing flushes as alternatives-but the pegging power of this hand is too great to pass up

   

Friday 5:05 AM

Went with the in crowd today.

   

Friday 5:06 AM

I'll deviate. I like the flush and like the 3-5 in the crib. I'll be in attack mode. I can only account for 13+1+2 points. Need to get myself close enough so a 10 or 12 hand will put me out

   

Friday 5:23 AM

I like come from behind wins

   

Friday 5:46 AM

3-4-5-6 flush with (5-7) to my crib.

Every cut is helpful to my hand, 5-7 is helped immediately by 333,55,666,777,8888 and 16*X.

   

Friday 5:53 AM

I don't normally break up the 4556, but tossing the 3-5 to the crib while holding the flush, simply intrigued my curiosity.

With 5 spades already accounted for, the percentages of another spade are reduced. Between the hand and crib I will need to find an additional 4-5 points by either the pegging or by the pone's contribution to the crib to reach my objective.

   

Friday 5:57 AM

Ras might be wagging his finger @ me later today for passing up “his“ flush....Hal too...time will tell

   

Friday 6:08 AM

Nice cut for me giving me 13 in hand and hoping for K- 10 in Crib

   

Friday 6:23 AM

Interesting choices today. Kept 4567 together for magic 11 & sweet 16. Yes, starting points are 2 less and no hope for double double, but thinking I can peg well and 5 in the crib will be good since no X cards showing in my hand.

   

Friday 6:47 AM

Flushes certainly intriguing, but needing 30 points we need to go for the 23 or 24 hand IMO. If the dreaded 2 gets flipped we have the 3 to go with it in the crib; the dreaded 8 will have the 7 to accompany it.

   

Friday 6:54 AM

I decided not to flush because the chance of loadsapoints from this double run.

   

Friday 7:04 AM

Great puzzle today!

   

Friday 10:43 AM

Straight and flush...almost everything hits...2 magic 11s...a 5 in my crib with a possible little run...

   

Friday 2:33 PM

There's two flushes and 4-5-5-6:

3-4-5S-6: 10pts + 6pts (Schell: 6.01) = 16pts

4-5S-6-7: 10pts + 6pts (Schell: 5.97) = 16pts

4-5-5-6: 12pts + 3½pts (Schell: 3.58) = 15½pts

Potential:

3-4-5S-6: Improves with AAAA, 2222, 333, 444, 55, 666, 777, 8888, 9999 + 16xXs = 46 cuts = 46/46 = 100.0% up to 13/14/18/20pts with 2222, 333, 444, 55, 666, 777 = 18 cuts.

4-5S-6-7: Improves with 2222, 333, 444, 55, 666, 777, 8888, 9999 + 16xXs = 42 cuts = 42/46 = 91.3% up to 13/16/18/20pts with 333, 444, 55, 666, 777, 8888 = 18 cuts.

4-5-5-6: Improves with AAAA, 333, 444, 55, 666, 777, 9999 + 16xXs = 38 cuts = 38/46 = 82.6% up to 16/23/24pts with 444, 55, 666 + 16xXs = 24 cuts.

Position:

We're 5pts short of position and Pone is 6pts short of where they would like to be. I think we need to prevent opponent pegging too many to make sure they don't peg out as Dealer next time. So I'll play Defense.

Pegging:

Playing Defense none of these hands are ideal.

Summary:

The flushes have the best starting value by ½pt over 4-5-5-6 and they have more cuts for improvement but 4-5-5-6 has 24 cuts for 16-24pts so I'll throw the 3-7.

   

Friday 2:39 PM

At 91*-105 playing an Offense strategy for the pegging the dynamic expected averages and Win/Loss %s are:

_______________Our
Offense__Hand__Pegs_Crib_Total____W1 %____W2 %
3-4-5S-6__13.89+3.83+5.79=23.51_____4.3____52.7
4-5S-6-7__13.59+3.85+5.69=23.13_____4.7____53.0
4-5-5-6___16.04+3.48+3.44=22.96____10.1____52.0

Offense______L1 %____L2 %
3-4-5S-6______12.7____41.9
4-5S-6-7______11.9____41.8
4-5-5-6_______11.4____40.3

3-4-5S-6 is best for expected averages by 0.38pt but 4-5-5-6 is best for Win %s by a significant margin and is slightly lowest for Loss %s. So as we're approaching a critical position I'll decide based on Win/Loss %s and select 3-7 to discard.

After the 8 cut I'll play Defense to the lead.

   

Friday 4:28 PM

Choosing an offense strategy from the outset and proceeding with optimal pegging strategy once seeing the starter card which adds nothing to hand and a known two points to crib (likely 5 points in crib with either 6 or 9-X discard). Even so, we could be surprised with 10 or more in crib as n/d 16 holes out is likely to keep best hand; so we just might pick up two mid-cards in the crib. Not likely that n/d can score 16 on this deal (1 of 9). So will play off the lead although would play the 5S on any X-point lead. Would likely play the 5S on A, 2, 8, 9 as well. Need to split the fives and retain three in a row. Might need to wiggle and avoid pegs or it might be possible for me to pick off a safe run of three. Just have to see how the pegging unfolds. If taking a run of three, don't want to give up a four-card run; so watch count carefully.

   

Friday 5:20 PM

Think I would try 5H6S. Get a five into the crib, keeping a 5, a straight and a flush in the hand. Not good enough a player to analyze, but feels like this maximizes hand and crib combo. Maybe 7-3 to crib is better than is seems to me.