Math problem

[Fluffy]

[hv=pc=n&s=s87hjt7654da654c5&n=sakjt9hd7caqj9743&d=w&v=0&b=8&a=1n2s(clubs)p3c(forced)p6cppp]266|200[/hv]

 

I had this math problem today, LHO leads ♥A (east plays ♥2), what is the best line?

 

1NT = 15-17.

[Hanoi5]

I ruff on the table, and play AK of spades, no Queen? I ruff a spade. I win if the Spade Queen is doubleton AND the King is onside (or with the doubleton holder) or spades are 3-3 and clubs 3-2. Finessing clubs seems to win only when the Spade Queen is doubleton and whenever the Club King is doubleton (onside).

[manudude03]

Finessing clubs seems to win only when the Spade Queen is doubleton or whenever the Club King is doubleton (onside).

 

FYP.

[gszes]

a maddening problem lho looks like a 100% favorite to hold the club K

(surely rho would x 6c after this bidding with kx(x) of clubs). The problem

then boils down to distribution of the club suit itself.

 

Assumptions from the bidding are as follows lho has 2 cards in each suit besides clubs

while rho also has certain minimums based on the max number of cards in each suit that

lho can hold. Rho has at least 2 spades 3 hearts and 4 diamonds. This means rho has only

4 free spaces for clubs vs lho having 7. It is even worse that this since we have a

practical limit in clubs essentially needing 32 so this further reduces the empty spaces

in both hands by 2 and rho now has only 2 spaces for the third club and lho has 5 spaces for

the third club. This means taking the club finesse is down to around 29%. This means we are

pretty much forced to rely on the 33 spade break good luck:))))))))))))))))))))))))))))))

[Mbodell]

Let's consider it single dummy with no bidding:

 

We are down if clubs are 5-0, and down if clubs are 4-1 unless the stiff is the K or T and that doesn't take a trick.

 

If we finesse in clubs we have:

 

0 losers on: all 2=3 with the K onside.

1 loser on: all other 3-2, also 1=4 with stiff K or T and 4=1 with stiff T.

2+ losers on: all other 4-1 and all 5-0.

 

With no knowledge of honors or shape from bidding that works out to around:

0 losers: 13.56%

1 losers: 62.72%

2+ losers: 23.69%

 

Looking at spades alone (and not considering the bidding or the split in clubs), and playing spades from the top we get around:

0 losers on: all stiff Q or 4-2 with Qx doubleton

1 loser on: everything else

 

That works out to:

0 losers: 18.57%

1 losers: 81.43%

 

So at a high level (i.e., not correcting for conditional probability or the auction) going after clubs first gives you:

13 tricks: 13.56% * 18.57% = 2.52%

12 tricks: 13.56% * 81.43% + 62.72% * 18.57% = 22.69%

11 or less: 23.69% + 62.72% * 81.43% = 74.76%

 

Now instead if we play spades from the top for the ruff and then go after clubs we get:

Spades play for:

0 losers: Q stiff, doubleton, or tripleton.

1 loser: spade in 4 or longer suit

 

which works out to:

0 losers: 54.09%

1 loser: 45.91%

 

If we have to play clubs from the top we have:

0 losers: never

1 loser: all 3-2 + all 4-1 with stiff K or T

2+ losers: all others

 

which works out to:

0 losers: 0%

1 loser: 79.10%

2+ losers: 20.90%

 

Combining these we get:

13 tricks: never

12 tricks: 54.09% * 79.10% = 42.79%

11 or less: 57.21%

 

However, if the Q of spades drops stiff or doubleton we can then switch to clubs. We might choose to play the finesse or not, however, we also now have to worry about a spade ruff. If the Q drops singelton or doubleton onside, we probably want to play clubs from the top to make less likely a spade ruff. If the spade drops singleton or doubleton offside, we can now take the club finesse since RHO will not be able to give LHO a ruff with no spades.

 

So now the spade situation is:

Case 1: Q or Qx with RHO = 9.29%

Case 2: Q or Qx with LHO = 9.29%

Case 3: Qxx with anyone = 35.52%

Case 4: Qxxx or Qxxxx or Qxxxxx with anyone = 45.90%

 

In case 4 we are dead, in case 1 we get the finesse clubs percentages we started with, in case 3 we get the clubs from the top just above, but in case 2 we also play clubs from the top, but we now lose on Kx offside due to the spade ruff available.

 

Combining this we get:

13 tricks: case 1 with 0 club losers (start) = 9.29% * 13.56% = 1.26%

12 tricks: case 1 with 1 club loser (start) + case 2 with 1 club loser (above - Kx offside) + case 3 with 1 club loser (above)

= 9.29% * 62.72% + 9.29% * (79.1% - 13.56%) + 35.52% * 79.1% = 40.01%

11 or fewer: 58.73%

 

So the extra flexibility of playing clubs differently if the spade Q is short lets us occasionally take all the tricks, but costs us slightly overall due to the increased ruff chance.

 

Now all of that was ignoring the bidding. It is unlikely, but not impossible, that the spade Q is stiff with LHO given the 1nt bid. The points suggest that many cards are also more likely to be with LHO, making the club finesse more likely to occur. But does that make up for the 41.28% chance playing spades first compared to the club finesse first which only wins 25.21% of the time?

 

If we assume LHO has AK of hearts at least, then there are 13 missing HCP and LHO needs 8-10 of the 13. There are 6 missing honor cards and thus 64 ways to arrange those honors between LHO and RHO, but only 19 of these ways give LHO between 8-10 points. And of those LHO has the CK 14 (the SQ 12). While everything isn't exactly equally likely that make the chances of the club K onside about 73.7%.

 

So 2=3 shape in clubs is about 33.9%. Before we were saying that Kx onside was 13.56% of that (4/10 of that). But now we are saying of the 10 2=3, the 4 with the K onside count for weight of around 3, the 6 with the K offside count for weight of about 1. This makes Kx onside be about 12/18 of 33.9 or 22.6%. Perhaps slightly more when we don't allow 0=5 or 1=4 shapes either, so call that 26.93%.

 

I don't think this will be enough to make playing on clubs better. Because we'll make at least 12 tricks this 26.93% as well as the roughly 50% of the time we have 1 club loser times the 18% of the times spades from the top come in and that will only get us to about 35-40%. Certainly better than the ~25% of the original calculation of this order of play, but not as good as the 41-odd percent of the spades first line from above. And the spade first line will go up a bit with the K likely to be onside, as we are less likely to get a ruff.

[whereagles]

The solution to this problem is:

 

Stay out of bad slams :)

[Fluffy]

I forgot to add that in Spain people do not open 1NT with 5 card major. And rarely with singletons, so spades won't be 5-1/1-5 and clubs won't be 1-4

[Mbodell]

Ok, so I decided to script this. I wasn't sure about the NT so I considered 3 different NT styles:

 

Strict NT where 2-4 S, 2-4 H, 2-5 D, 2-5 C

no 5M but 6m ok NT where 2-4 S, 2-4 H, 2-6 D, 2-6 C

loose NT where 2-5 S, 2-5 H, 2-6 D, 2-6 C

 

In the last one I allowed 5M422 hands to be opened 1nt, could restrict that too, but shouldn't materially effect the which line is best or the rough percentages.

 

I simulated 1,000,000 hands (with N and S fixed, W fixed with H AK, E fixed with H 2) and figured out which lines work with what percentage.

 

There were 1000000 hands dealt

There were 188643 inclusive NT hands dealt (18.8643%)

There were 151460 6 minor but no 5 major NT hands dealt (15.146%)

There were 144720 srtict balanced no 5 major NT hands dealt (14.472%)

There were 422712 times line1 - spades first - worked (42.2712%)

There were 260554 times line2 - club finesse first - worked (26.055400000000002%)

With a strict NT shape, line 1 - spades first - works (53.65%) while line 2 - club finesse first - works (39.26%)

With a 6 minor ok but 5M not NT shape, line 1 - spades first - works (53.68%) while line 2 - club finesse first - works (40.97%)

With a normal NT shape, line 1 - spades first - works (53.44%) while line 2 - club finesse first - works (45.43%)

 

If you ignore the odd floating point quirk there we see that when we ignore the bidding restrictions on shape and points that the spades first line works 42.3% compared to 26.1% for the club line. That is very close to the 42.8% and 25.2% that I had calculated in my Q&D by hand calculations.

 

If you restrict the W hand by HCP and shapes you see the odds of making go up, but the spades first line is always around 53.5%, the club finesse line depends on how strict or loose you are on the shapes and varies from around 39-45% or so. That is slightly better for each line than I figured in my Q&D calculations including points and shape (I thought it was going to be 45-50% for the spades first and 35-40% for the club finesse).

 

So interesting to see that given the bidding and opening lead and what that implies, we are actually in a marginally good slam - so long as we play on spades first.

 

Anyone that wants to check and validate my script code can see that here

[Fluffy]

Thanks a lot, I was neededing this.

 

If a suit can only break 4-2 or 3-3, it is 60% of the queen falling doubleton or tripleton. The 53% should be because of 4-1 clubs. I wonder why it drops so little when you allow 5 card major.

 

 

 

A rule of thumb could go like this: Club finesse plays for West to have a specific lenght: 2 and a card: ♣K

Spade finesse plays for West for a specific length: 3 and a card: ♠Q

 

But spade has 2 winning cases: when either player has ♠Qxx. 2 is usually better than one. Even when ♠Q offside is not likelly.

[Fluffy]

I looked at the code, and I think there is a small thing missing: When west has ♠Qx and East has ♣Kx we go down on a spade ruff.

[Mbodell]

I looked at the code, and I think there is a small thing missing: When west has ♠Qx and East has ♣Kx we go down on a spade ruff.

 

That is handled by the following lines:

 

if {$club_loser_top == 1} {

# Still need to avoid the ruff

if {!($clubs_west == 3 && [east has KC])} {

set line1 1

}

}

 

Note that this is in line 1, when West has 1 or 2 spades and the Q (obviously in the NT openers this will just be 2 spades - but I keep track of both for the overall count). If West has 3 clubs and East the CK we go down.

 

There is an extremely unlikely line that isn't covered: When spades are 3-3 and East has the SQ and the stiff K of clubs. If East plays the spade Q on the second round of spades, we'll think the spades are split 4-2 and we'll go over in diamonds and take a club finesse. We'll now lose 2 club tricks. If east doesn't find this play, we'll ruff the third round of spades and play clubs from the top and only lose the one club. But it is quite unlikely that East has Qxx of spades and stiff K of clubs, given west is the one with the points (but AKQ of H and KQJ of D is still 15 points). It is also quite unlikely that East finds that play. But what a defense if she does!

[Fluffy]

I've been there:

 

-Partner, are you void in clubs?

-What? oh sorry this is a spade. Sorry, I have a club here.

Later on...

 

Sorry partner, I had a spade between my clubs, surely if I had it right we would put 2 down.

[suokko]

Sorry to crash the simulation party a bit late. But here is my take on simulations with a bit different per cents to success but same line winning.

 

The simulation code that I used for exhaust mode. Exhaust mode generates all possible remaining hands when two hands are known.

 

predeal north SAKJT9,D7,CAQJ9743
predeal south S87,HJT7654,DA654,C5
predeal west HA
predeal east H2

westHCP = hcp(west) > 14 && hcp(west) < 18
west1NTstrict = shape(west, any 4333 + any 4432 + any 5332 - 5xxx - x5xx) && westHCP
west1NTno5M = shape(west, any 4333 + any 4432 + any 5332 + any 6322 + any 5422 - 5xxx - x5xx - 6xxx - x6xx - 22xx) && westHCP
west1NT = shape(west, any 4333 + any 4432 + any 6322 + any 5422 - 5xxx - x5xx - 6xxx - x6xx - 22xx + any 5332) && westHCP


westhandtype = west1NTstrict ?
(west1NTno5M ? 3 : 1) :
(west1NTno5M ? 2 : 0)

dropQxwest = spades(west) == 2 && hascard(west, QS)
dropQxeast = spades(east) == 2 && hascard(east, QS)
dropQeast = spades(east) == 1 && hascard(east, QS)
spaderuff = spades(west) == 3
spadefinesse = spades(west) <= 3 && hascard(west, QS)

clubfinesse = clubs(west) == 2 && hascard(west, KC)
clubsfinesseoneloser = clubs(west) < 4 || hascard(east, TC)

clubsoneloser = clubs(west) < 4 || hascard(east, KC) || hascard(east, TC)

cKonside = hascard(west, KC)
sQonside = hascard(west, QS)

dropQsinglethenclubs = dropQeast && (clubfinesse || clubsfinesseoneloser)

# Line A is club finesse (test SQ single then club finesse)
lineA = dropQsinglethenclubs ||
clubfinesse ||
(clubsfinesseoneloser && (dropQxwest || dropQxeast))

# Line B is spade finesse (test SQ single then clubs from top)
lineB = dropQsinglethenclubs ||
(spadefinesse && clubsoneloser)

# Line C is spade ruff (test SQ doubleton then ruff spade)
lineC = dropQsinglethenclubs ||
(dropQxeast && (clubfinesse || clubsfinesseoneloser)) ||
(dropQxwest && (clubsoneloser && (clubs(east) < 3 || hascard(west, KC)))) ||
(spaderuff && clubsoneloser)

condition west1NTstrict
condition west1NTno5M
condition west1NT

action
#	frequency "west hand type 3 = strict and no 5M, 2 = no 5M" (westhandtype,2,3),
average "♠Q onside" sQonside,
average "♣K onside" cKonside,
average "Club finesse" lineA,
average "Spade finesse" lineB,
average "Spade ruff" lineC,

 

 

Results:

condition west1NTstrict meaning only 4333 4432 and 5332 shapes allowed with no 5M
♠Q onside: 0.704208
♣K onside: 0.889556
Club finesse: 0.369362
Spade finesse: 0.421982
Spade ruff: 0.580242
Generated 2704156 hands
Produced 224820 hands
Initial random seed 1422703180
Time needed    0.278 sec

condition west1NTno5M meaning all balanced and semibalanced shapes without 5M or 22M
♠Q onside: 0.686543
♣K onside: 0.872162
Club finesse: 0.442801
Spade finesse: 0.425987
Spade ruff: 0.548297
Generated 2704156 hands
Produced 262285 hands
Initial random seed 1422703358
Time needed    0.323 sec

condition west1NT adding 5M332 hands to the previous simulation
♠Q onside: 0.68623
♣K onside: 0.873573
Club finesse: 0.451725
Spade finesse: 0.435276
Spade ruff: 0.560902
Generated 2704156 hands
Produced 294843 hands
Initial random seed 1422703487
Time needed    0.329 sec

[Mbodell]

Sorry to crash the simulation party a bit late. But here is my take on simulations with a bit different per cents to success but same line winning.

 

I assumed that West had the ♥K to go with the A for their opening lead. You did not. That likely accounts for much of the difference.

[suokko]

I assumed that West had the ♥K to go with the A for their opening lead. You did not. That likely accounts for much of the difference.

 

True. I would assume A lead to slam must be AK. But I probably should prevent DKQJ if no HK. But even with that opening leader might try to take A if thinking that there is sure trump trick. That why I didn't assume HK has to be with opening leader. Some lead agreements even make A lead to 5+ levels deny king. Too bad that kind of information is rarely provided in posts.

[Mbodell]

The simulation code that I used for exhaust mode. Exhaust mode generates all possible remaining hands when two hands are known.

 

I'm impressed that your exhaust mode solves all possible hands in less than a second.

[suokko]

I'm impressed that your exhaust mode solves all possible hands in less than a second.

 

It uses bit vector permutations to select how to distribute unknown cards for each hand. Bit vector is enough because only two hands can hold remaining unknown cards when two hands are completely known. That vector is then converted to bit hand presentation with bitwise operations. That part is about 10 times faster than generating a random deal using rng.

 

The remaining works is done by the script interpreted that is slow for complex scripts. In this case relaxing 1NT requirements produces more matching deals for analyze steps showing how much analyze scripts takes even in this fairly trivial looking case.

 

If no cards are known for remaining hands then there is 10 million possible deals to generate. Even that takes just under a second if using a very simple script.

[lamford]

(surely rho would x 6c after this bidding with kx(x) of clubs).

Why on earth would he do that? He has no intention of helping you with the play.

[gszes]

Why on earth would he do that? He has no intention of helping you with the play.

 

:)))) after this bidding I would sincerely expect my p to rise from the table leave

for the bar and never return if I could not find x with kx(x) on this bidding. I will

gladly help declarer a bit with their play on hands where they should have no chance

whatsoever and will keep yet another of my limited supply of cherished partners:)

[gszes]

We have some highly talented programmers here but I have a question ==

what kind of calculations and why would you use to solve this problem ----

at the table ? If you can do these types of calculations in your head that is

great or else it seems to more of a case of trivial pursuit rather than any sort

of useful technique..

I realize there is genius out there but our game is supposed to be mental not

based on the ability to use a calculator or type programs faster than anyone else.

Localizing the problem down to determining if the club finesse or depending on 33

spades or the Q falling doubleton is mental try doing the rest that way its sort of

fun though frustrating at times.

[Mbodell]

We have some highly talented programmers here but I have a question ==

what kind of calculations and why would you use to solve this problem ----

at the table ? If you can do these types of calculations in your head that is

great or else it seems to more of a case of trivial pursuit rather than any sort

of useful technique..

I realize there is genius out there but our game is supposed to be mental not

based on the ability to use a calculator or type programs faster than anyone else.

Localizing the problem down to determining if the club finesse or depending on 33

spades or the Q falling doubleton is mental try doing the rest that way its sort of

fun though frustrating at times.

 

I think you can get better at doing these mentally if you practice them offline. You obviously can take shortcuts much of the time rounding things off a lot since many lines aren't that close.

[Lovera]

I don't know how much is it good but i have quickly thinked that if spade are 3-3 with Q offside (2 points) ruffing third round and after running club (K offside 3 points than 5 in E) with A of diamond we could have 12 tricks.

[shellsnail]

At the table I will ruff, cash ace club, play queen club. finesse the spade queen, draw trump and claim 12 tricks.

 

caters for king club singleton offside + any break.

caters for Qxx, Qx spade onside. + any trump 3-2 or better.

 

The alternative of trying to finesse clubs only works with 4-2 spades q spade dropping or Kx club doubleton onside. I'm not doing any calculations at the table but intuitively this seems like a worse line. What's the verdict?