Chuck Arthur

The Wednesday Game

The Wednesday Game (you may click on that name to visit the site) is both the name of a website and a game held the first Wednesday of each month. There is nothing ACBL-wise official about this game; it costs no extra to play in it and no charity or international fund benefits from it. Players at participating clubs throughout North America  (perhaps the whole world, as far as I know) play a set of pre-dealt hands. These are not specially arranged: simply random computer dealt hands. Prior to the first Wednesday, the guru at that website emails materials to the participating clubs to allow them to pre-deal the hands. After the game, participants may visit the website where the hands are analyzed. During that Wednesday, one requires a password (given to the players at the completion of the game) to access the analyses for the current month, until play is complete at clubs on the west coast. There is a wealth of additional information at the website; for instance there is what he calls a BridgeOpedia where he keeps many articles on various topics in bridge.


Dealer: West

Vul: EW

Deal: 16























West North East South
1 Pass 3 Pass
4nt Pass 5 Pass 
6   All Pass    

This was one of the hands this month. I was West. I did not think that my hand quite met the standards for an almost-forcing-to-game 2 clubs, so I opened it 1 heart. 3 hearts was a limit raise; 4 notrump was RKCB; 5 hearts showed 2 keycards and denied the Queen of trump. I knew that we had all the keycards but that we were off the Queen of trump, so I checked out at 6 hearts.

The opening lead was the King of spades. I won and drew two rounds of trump, discovering that I had a trump loser. Suddenly, my chances of making this slam had decreased dramatically. At first glance, it seems that the club loser is unavoidable, but I had what I thought was a pretty good plan. I cashed my diamond winners, ruffing dummy’s last spade along the way, and cashed one high club. Then I threw in North with her trump winner. If she holds the Queen of clubs along with her trump winner, she is now endplayed. Alas, it was not to be; she had an easy exit in the 10 of clubs and I went one down. This line of play works whenever North has the Queen of clubs (50%) plus some extra chances: singleton Queen, Qxxx, or Qxxxx with South. I make these extra chances to be 5/10*28.26%  + 1/2*3.91% = 16.08%. Altogether that is a 66.08% chance of success.

Perhaps that was a good plan, but not good enough. Two declarers, John Cunningham and Steve Gittens were in slam and made it. They followed a line of play similar to mine up until the point where I cashed one hign club. They cashed TWO high clubs before throwing North in with their trump winner. Now North was truly endplayed and the slam came home. That line of play works whenever South has the Queen of clubs any number of times (again 50%, but a different 50) plus some extra chances: North having the singleton Queen or Qx of clubs. I make those extra chances to be 16.39%. Altogether, that is a 66.39% chance of success. I was bitten because I took a line of play that was 0.31% inferior. I am simply not good enough to work out those probabilities in my head at the table. My hat is off to John and Steve who, I guess, were.

You may see the virtual traveller here.


Steve GittinsNovember 5th, 2010 at 11:01 am

Actually, Chuck, when North shows in to all four rounds of diamonds the odd have changed dramatically. He is known to have three hearts and four diamonds. So if he has four or more clubs south would have eight plus spades and would surely have bid four spades white against red. So the only layout my line of play loses to is Qxx in the North which is 81.25%

john cunninghamNovember 5th, 2010 at 11:19 am

It is important to ruff out the spade prior to attempting to run the diamonds, of course.

Having done that,

If you cash only one club, then run the rest of diamonds, you are not necessarily out of the woods with an eventual forced club from lho.

sometimes he has 10,9,8 tripleton (a possibility against an imaginitive player)

It is interesting that not having the club eight affects the order of play, perhaps. With the eight, I would be more comfortable running diamonds before attempting a second club. Probably doesn’t matter, though it gave me pause.

Steve MackayNovember 5th, 2010 at 11:54 am

Hi Chuck,

I hate to say this but are you (and the others) not making this way too complicated? Surely a grade three kid would adopt the winning line and I doubt if it would be after complicated mathematical calculations. For the line taken by Steve and John to work (including, as John says, first ruffing the spade), all that is required is that S have more clubs than N.

N has three hearts. S has one. Need we discuss it any further?

No guarantees, but if you kept making bets like this at Las Vegas, you (and the grade three kid) would be rich in no time.

Steve MackayNovember 5th, 2010 at 12:09 pm

Hi again Chuck,

My entry to your blog was not complete. I should have added that, after ruffing the spade, if the diamonds lived, cashing both high clubs before exiting in trumps was marked.

So yes, there is a decision – whether to try to cash the fourth diamond before or after cashing the second high club.

Steve GittinsNovember 5th, 2010 at 12:24 pm

Hi, Again,

Responding to Steve M’s second note, I think it’s clear to cash the fourth diamond before cashing the second club. If North shows in we’re back to my 81% line and if he ruffs it we put up the J of clubs when he exits a club a la Chuck’s 66% line.

lewis richardsonNovember 5th, 2010 at 5:24 pm

with south having the singleton heart, he is odds on to be the one with longer clubs. i thinkk the odds shift dramatically to that conclusion. of course it makes perfect sense to strip the hand first.

Chuck ArthurNovember 5th, 2010 at 8:23 pm

John, Steve, and Lew, you make good points.

Tim CapesNovember 5th, 2010 at 9:35 pm

Lead/Inferential Count:

North has 3+S (unless they played the QS on the spade ruff), 4D and 3H so they have 3- clubs.

If south is the type to come in with a 7 card suit white-red, then you can be fairly sure they have 4S which makes cashing the AK and throwing N in 100%.

Cashing the AK will work whenever North has 2 clubs (any) or whenever North has 3 small clubs (queen falls).

Cashing the A and playing another club works whenever North has QX in clubs or whenever North has QXX in clubs (North always has 2 or 3 clubs on the count). You now lose when N has XX or XXX in clubs.

So it becomes a matter of winning QXX opposite XX

losing XX opposite QXX

losing XXX opposite QX

The first two holdings cancel out, the 3rd holding is your edge, and your edge is further enhanced by a belief that N has shortness making 2 clubs more likely than 3.

Chuck ArthurNovember 6th, 2010 at 9:35 am

As declarer is about to play to trick 10, he must decide who is more likely to have the Queen of clubs. If it is South, he must cash the other high club before putting North in with her last trump. If it is North (unless she started with a doubleton, in which case it doesn’t matter), he must throw her in right now. The inference is that North still has the Queen of spades: I’m going to give it to her. It is proven that she has the Queen of hearts. There are 2 empty spaces in North, 4 in South. It is 2:1 to play South for the Queen of clubs.

As an aside to the play of the hand… as declarer is ruffing out the spade, North should play the Queen, in an attempt to obfuscate declarer’s count of the hand.

In addition to those who commented above, Harold Baba also responded to me privately. My thanks to all respondents.

LindaNovember 9th, 2010 at 11:14 am

Great blog. We are adding this to our theme about reviewing websites.

Jim de BoerNovember 12th, 2010 at 7:03 pm

This hand fascinated me, so I spent some time creating an EXCEL spreadsheet to figure out the best line of play.

Stealing from everyone else’s ideas, the first 5 tricks are:

1.) Win A of spades.

2.) A of hearts

3.) K of hearts, south discarding a spade.

4.) Diamond to Ace, both follow.

5.) Ruff a S – both follow low (but as Chuck says, N should play Q; let’s say he didn’t).

North is known to still have QH and QS. So he started with 3, 4, or 5 spades. North would have bid over 1 heart with 6 spades to KQ, and Q of hearts. So south has 5, 6, or 7 spades.

With this basis, here are the best lines of play.

Best line (75.77%) is cashing 2nd D, then A C, then 3rd and 4th D.

Next best (75.22%) is cashing 2nd & 3rd D, then A C, then 4th D.

Next best (74.10%) is cashing 2nd, 3rd, and 4th D.

Next best (68.52%) is cashing 2nd D, A C, 3rd D, K C, 4th D.

Next best (67.97%) is cashing 2nd & 3rd D, A & K C, 4th D

In all lines, adjust the play once you know North’s count of diamonds:

When south shows out on 2nd D you’re home free: (north has 6 D, therefore 0 or 1 C), play 3rd & 4th D, A & K of clubs, then play H to N.

When south shows out on 3rd D, you’re home free: (north has 5 D, therefore 0, 1, or 2 C), play 4th D, play A & K of clubs, then play H to N.

When south shows out on 4th D (north has 4 D), N most likely has 2 C (22%) or 1 C (5.6%); vs 3 C (11%), so, play A & K of clubs, then play H to N.

Play J on low club lead from north.

If the Q C falls on A C, then you’re home free.

Here are all the possible distributions for north, with probabilities.

Note: the probabilities were calculated using a random number generator which dealt out the 12 unknown cards; 9000 samples; so it is not perfect, but close.

5=3=5=0 0.50%

5=3=4=1 5.59%

5=3=3=2 10.13%

5=3=2=3 5.18%

5=3=1=4 0.68%

4=3=6=0 0.21%

4=3=5=1 5.24%

4=3=4=2 22.08%

4=3=3=3 21.87%

4=3=2=4 5.27%

4=3=1=5 0.18%

3=3=6=1 0.58%

3=3=5=2 5.12%

3=3=4=3 11.16%

3=3=3=4 5.61%

3=3=2=5 0.61%

Note that the total probability that north has more clubs than south, before playing the 6th trick, is 50.54%. This is because south is known to have the same or more spades than north: 5-5, 6-4, or 7-3.

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