The Schnapsen Log

Homework on Expected Values

2013-04-26T11:00:00-08:00

The Schnapsen Log

April 26, 2013

Homework on Expected Values

Martin Tompa

For regular readers of this column, I apologize for the fact that I have been producing columns at a slower rate recently. The slow rate is probably going to continue for the next few months. I haven’t been idle on the Schnapsen front, however. I am busy teaching a course on Probability and Statistics, and am using Schnapsen as a running example of applications of Probability. It’s been fun teaching Schnapsen to a large group of students who had never encountered it before.

In the course, we are just up to the topic of expected value now, so it seems appropriate to give them a homework exercise that involves expected game points. Today’s column is that homework exercise. This means that I won’t be posting my analysis until one week from now, when the homework will be due.

You have just started a new game against the Maestro. On the very first deal, you reach the following interesting position:

Unseen cards:
♠ Q
♥ KJ
♣ TK
♦ A

Your cards:
♠ A
♥ ATQ
♣ —
♦ T

Trump: ♠J
Stock: 1 face-down card
Game points: Maestro 7, You 7
Trick points: Maestro 21, You 18
On lead: Maestro

The Maestro fingers each card in his hand in turn, and finally leads ♦A. There have been no marriages declared, and no one did a trump exchange. You should assume that each of the five cards you haven’t seen is equally likely to be the last face-down card in the stock.

Plan your play for the rest of the hand. In particular, answer the following questions:

(a) How will the deal play out if you duck this trick? Who will win, and how many game points?

(b) How will the deal play out if you win this trick? In this case, you will draw the random, face-down card from the stock and the Maestro will draw the face-up ♠J. Consider each of the five cards you might draw and, for each one, find your best play and the resulting number of game points you will win or lose. Combine these appropriately to determine the expected number of game points that you will win.

Once your homework has been turned in a week from now, you will be able to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Rags and Riches

2013-04-13T11:00:00-08:00

The Schnapsen Log

April 13, 2013

Rags and Riches

Martin Tompa

You are playing your favorite game against your favorite buddy Peter in your favorite hangout, the Café Abeles. You’ve recouped some of your losses, coming from a depressing game point score of 1:6 a few deals earlier to your present score of 1:2. Can you possibly continue making the right decisions and go on to win this close game?

Unseen cards:
♠ ATJ
♥ —
♣ TK
♦ A

Your cards:
♠ —
♥ TQJ
♣ AQ
♦ —

Trump: ♣J
Stock: 1 face-down card
Game points: Peter 1, You 2
Trick points: Peter 26, You 15
On lead: You

You have only 15 trick points and are staring at some more ragged looking cards in your hand. The majority of aces and tens appear to be in Peter’s hand. What should you do at this point? When you think you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

How Much Do You Trust to Chance?

2013-03-27T11:00:00-08:00

The Schnapsen Log

March 27, 2013

How Much Do You Trust to Chance?

Martin Tompa

The battle with your buddy Peter continues. It would be arduous if it weren’t for the Café Abeles’s good supply of coffee and cake.

Unseen cards:
♠ ATQ
♥ TQ
♣ T
♦ —

Your cards:
♠ J
♥ A
♣ AJ
♦ Q

Trump: ♥J
Stock: 1 face-down card
Game points: Peter 1, You 4
Trick points: Peter 12, You 50
On lead: You

Now this is more to your liking. You picked up 2 game points in the previous deal, and in this deal you’re far ahead in trick points, thanks to the diamond marriage you declared earlier. How do you proceed? When you think you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

A Multitude of Options

2013-03-12T11:00:00-08:00

The Schnapsen Log

March 12, 2013

A Multitude of Options

Martin Tompa

Back by the roaring fire in the Café Abeles, you’re on to the next deal. Your last desperation play didn’t succeed, as you might have expected. Your good buddy Peter won 2 game points and is now just 1 game point from winning.

Unseen cards:
♠ TQ
♥ J
♣ AKJ
♦ —

Your cards:
♠ AK
♥ A
♣ TQ
♦ —

Trump: ♦J
Stock: 1 face-down card
Game points: Peter 1, You 6
Trick points: Peter 20, You 27
On lead: You

There’s no need to lose hope just because you’re this far behind in game points. All it takes to catch up are a few lucky deals. This one is unusual, because all the trumps, except for that one that’s face-up on the table, are already gone.

There seem to be lots of interesting options to consider. Which one looks best to you? When you think you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

To the Brave Go the Spoils

2013-03-07T11:00:00-08:00

The Schnapsen Log

March 7, 2013

To the Brave Go the Spoils

Martin Tompa

A warm fire, a piece of Linzertorte, coffee capped with whipped cream, a good friend across the table, and cards. Does life get any better?

“Hey, daydreamer,” Peter interrupts your pleasant thoughts. “It’s your lead.” You look down and recall the position you’re in:

Unseen cards:
♠ AK
♥ K
♣ AKQ
♦ —

Your cards:
♠ TJ
♥ T
♣ —
♦ TJ

Trump: ♠Q
Stock: 1 face-down card
Game points: Peter 3, You 6
Trick points: Peter 37, You 29
On lead: You

This is one of those situations when being on lead is awkward. What’s your plan? When you think you have a good one, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Where's the Endplay?

2013-02-26T11:00:00-08:00

The Schnapsen Log

February 26, 2013

Where’s the Endplay?

Martin Tompa

In the Café Abeles on this cold winter day, you and Peter took the opportunity of a break between games to replenish your supplies of hot coffee and cake. This time you have given in to your weakness, Linzertorte, a nutty-flavored cake with raspberry filling and a beautiful lattice crust on top. It’s difficult to savor it when you can’t stop yourself from loading the next piece onto your fork. The taste brings back fond memories of your grandmother, who used to bake a very similar cake.

While you were thus distracted, Peter won the first few deals handily. Finally looking up from your plate as though awakening from a sweet dream, you find yourself in the following position:

Peter: (34 points)
♠ AQ
♥ —
♣ J
♦ AK

You: (47 points)
♠ TJ
♥ —
♣ TQ
♦ Q

Trump: ♣
Stock: Exhausted
Game points: Peter 3, You 7
On lead: You

Peter declared a marriage earlier in this deal. You eye your spade holding with some displeasure. It looks to you as though there must be some endplay here, but it keeps slipping from your grasp. Plan your play. When you think you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

An Unusual Capture

2013-02-20T11:00:00-08:00

The Schnapsen Log

February 20, 2013

An Unusual Capture

Martin Tompa

The first game against your buddy Peter in the warmth and comfort of the Café Abeles is nearing its conclusion. You find yourself on lead in the following position:

Unseen cards:
♠ K
♥ TKJ
♣ A
♦ J

Your cards:
♠ A
♥ AQ
♣ K
♦ A

Trump: ♦K
Stock: 1 face-down card
Game points: Peter 4, You 2
Trick points: Peter 20, You 23
On lead: You

What do you think of your prospects this time? Plan your play. When you think you have arrived at a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

On a Wing and a Prayer

2013-02-16T11:00:00-08:00

The Schnapsen Log

February 16, 2013

On a Wing and a Prayer

Martin Tompa

You are wrapped in the warmth of the Café Abeles, on your second steaming cup of Kaffee mit Schlag, playing against your good buddy Peter. Your have only one care in the world: playing you cards in the best order possible. Here is your position:

Unseen cards:
♠ KJ
♥ AQ
♣ —
♦ KQ

Your cards:
♠ —
♥ J
♣ ATK
♦ A

Trump: ♦J
Stock: 1 face-down card
Game points: Peter 5, You 2
Trick points: Peter 46, You 47
On lead: You

Both you and Peter are within striking distance of 66. What do you think of your prospects? Plan your play. When you think you have the best plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

A Bird in Hand

2013-02-11T11:00:00-08:00

The Schnapsen Log

February 11, 2013

A Bird in Hand

Martin Tompa

Warmed by the blazing fire in the Café Abeles, you shuffle and deal out the next hand. The first few tricks go uneventfully, with no marriages and no trump exchange. At the beginning of trick 5 you reach this position:

Unseen cards:
♠ Q
♥ T
♣ K
♦ ATK

Your cards:
♠ T
♥ J
♣ QJ
♦ J

Trump: ♥A
Stock: 1 face-down card
Game points: Peter 5, You 3
Trick points: Peter 27, You 21
On lead: Peter

Peter gets a knowing look in his eyes and plunks ♦A down on the table. You inspect the meager collection of small cards in your hand and face an unhappy choice: do you want Peter’s ace in your tricks or do you want to pick up the ace of trumps for your hand? Make your choice. Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Imperfect Information

2013-02-05T11:00:00-08:00

The Schnapsen Log

February 5, 2013

Imperfect Information

Martin Tompa

In the Café Abeles with your good friend Peter on this wintry day, you look a little sadly at your dwindling piece of Haselnusstorte and your cup of coffee, which not even an optimist would call half full. “Ah well”, you think, “I’m sure they have more in the kitchen.” Meanwhile, you are only a few tricks into the next deal when you find yourself in this situation:

Unseen cards:
♠ —
♥ TQ
♣ ATK
♦ TKJ

Your cards:
♠ AK
♥ AK
♣ Q
♦ —

Trump: ♥J
Stock: 3 face-down cards
Game points: Peter 5, You 5
Trick points: Peter 13, You 38
On lead: You

We haven’t looked at too many positions this early in the deal. When you are at trick 6, you have perfect information about your opponent’s hand, assuming you’ve remembered the played cards correctly. When you are at trick 5, you have nearly perfect information about your opponent’s cards, since there is only one card concealed in the stock and the other 5 are held by your opponent. But at trick 4, where you find yourself now, your information is quite incomplete: nearly half the cards you haven’t yet seen are still in the stock.

How should you proceed from this position? Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Danger at Every Turn

2013-01-29T11:00:00-08:00

The Schnapsen Log

January 29, 2013

Danger at Every Turn

Martin Tompa

You have finally warmed up by the roaring fire in the Café Abeles. The smell of wood smoke fills your nose, making you realize how much you enjoy the comforts you have on this cold winter day. Your coffee has cooled enough to sip after each small bite of the café’s justly famous Haselnusstorte. Having appreciated these creature comforts, you turn your attention back to the problem at hand.

Unseen cards:
♠ TQ
♥ TQ
♣ —
♦ KJ

Your cards:
♠ AJ
♥ AJ
♣ A
♦ —

Trump: ♥K
Stock: 1 face-down card
Game points: Peter 5, You 6
Trick points: Peter 54, You 33
On lead: You

You have each declared a marriage, yours in clubs and Peter’s in diamonds. You now exchange your ♥J for the ♥K showing on the table. That was the easy part. Now what? Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Schemes for Remembering Cards

2013-01-28T11:00:00-08:00

The Schnapsen Log

January 28, 2013

Schemes for Remembering Cards in Schnapsen

Martin Tompa

One of our readers asked for the details of a method for remembering which cards have been played and which still remain unseen. For beginning and intermediate Schnapsen players, I know this memory work seems like a daunting prospect. Yet it is also the most essential skill to master because, without knowing which cards remain unseen, it is impossible to plan the endgame. We’ve been learning about sophisticated endplays such as eliminations, squeezes, and counterforce plays, but you won’t be able to pull them off at the Schnapsen table unless you can visualize your opponent’s hand.

I have presented some basic ideas that I use for remembering the cards in my article Winning Strategy for Schnapsen. Included there are some pointers about how to practice increasing your card memory. An example is to start out just trying to remember which trumps have been played so far, without trying to remember the other three suits.

But today’s column isn’t about how to practice, it’s about detailed schemes for keeping the cards straight in your mind. One device that I find helps me manage all these cards is that I never remember more than 2 cards per suit. At first I remember the cards that have been played, but as soon as I have seen at least 3 in the suit (including any in my hand), I switch over to remembering the cards I haven’t yet seen. Since each suit has only 5 cards, I never have to remember more than 2. A second benefit of this method is that, by the time I get to trick 5 or so, I’ve completely switched over to remembering all the cards that I haven’t seen, which is exactly the view needed to plan out the endgame.

Here are more details of how I’m tracking these cards in my head. After each trick, I recite to myself the list of 1-2 cards per suit, either those that were played or those I haven’t yet seen. I find that I don’t need to include the trump suit in this recitation: I can remember the trumps without explicit repetition. As an example, consider the following hand at the end of trick 2:

My cards:
♠ AK
♥ —
♣ K
♦ TQ

Trump: ♥J

Suppose that, in trick 1, ♦A was trumped with ♥T and, in trick 2, ♦J was led and ♣Q was discarded. What I would recite to myself at this point is, “King of diamonds, Queen of clubs”. The meaning of this to me is that the ♦K is the only unseen diamond, and ♣Q has been played. I always list first the suits in which I’m reciting the unseen cards, followed by the suits in which I’m reciting the played cards. Notice that I don’t include ♥T in my recitation, because it’s a trump. The trumps are important enough that I can remember them without explicit recitation. Another thing that I remember implicitly is the “border” between the unseen-card suits and the played-card suits in my recitation. In this example, that border happens between diamonds and clubs. Any suit in which no card has been played yet, or in which I’ve seen all five cards, is simply omitted from the recitation. Again, I have no trouble remembering whether such an omitted suit is untouched or exhausted.

Continuing the example, suppose that in the next trick my opponent leads ♠J, I take it with ♠K, and I draw a trump from the stock. What I would then recite to myself is, “King of diamonds, Ten Queen of spades, Queen of clubs”. Notice that I insert the spade suit before the club suit, because the spades I’m listing are cards that remain unseen and the club I’m listing is a card that was played. The border between unseen-card suits and played-card suits now happens between spades and clubs. Notice also that, in suits such as spades where there are equal numbers of played and unseen cards, I list the unseen cards. There is a slant toward the unseen-card view, since this is the view I will need during the endgame.

It probably sounds more complicated than I think it really is. This scheme evolved as I was training my own memory, and I think it probably evolved so as to minimize the number of cards I had to recite. As part of that evolution, I discovered at some point that I didn’t need to include the trump suit. I wish that I could just remember all the suits the way I remember the trump suit, without the need for explicit recitation, but I don’t seem to be there yet.

I have a friend whose method is somewhat different from mine, and perhaps it will be helpful to some readers to share his alternative method. Like me, he recites cards in the nontrump suits and recites at most two cards per suit, either those already played or those still unseen. But, unlike me, he always recites the suits in the same fixed order: ♦, ♣, ♥, ♠, omitting whichever suit is trump. Because he uses this fixed suit order, he also adds the word “no” to distinguish those suits in which he is listing cards that have been played from those suits in which he is listing cards still unseen. If there is a suit that hasn’t yet been played, he uses the keyword “all” in that suit’s position in the recitation. If there is a suit in which no unseen cards remain, he uses the keyword “none”.

Let’s go through the small example again, using my friend’s recitation scheme.

My friend’s cards:
♠ AK
♥ —
♣ K
♦ TQ

Trump: ♥J

Remember that, in trick 1, ♦A was trumped with ♥T and, in trick 2, ♦J was led and ♣Q was discarded. What my friend would recite at this point is, “King, no Queen, all”, meaning ♦K is unseen, ♣Q has been played, and all spades are unplayed. Notice that the trump suit (♥) is omitted from the recitation. Notice also the slant toward unseen cards: “no Queen” in this example means that ♣Q is not among the unseen cards.

Continuing the example, suppose that in the next trick the opponent leads ♠J, my friend takes it with ♠K, and he draws a trump from the stock. What he would then recite is, “King, no Queen, Ten Queen”, meaning that now ♠TQ are the unseen spades.

What I like about my friend’s method is that he doesn’t have to recite the suit names; they are implicit in the order of his list. I also like the fact that certain information is explicit in his recitation and doesn’t rely on implicit memory, such as whether the cards listed are played or unseen. I find it interesting that he has chosen different items to leave implicit (the suit corresponding to each list of cards) than I have (which lists are played cards and which are unseen cards).

Finally, I have to make a confession about all this memory work. My friend and I both play mostly against two computer opponents, Master Schnapsen/66 and Doktor Schnaps. Both of these programs display the trick point totals of each player, so you don’t have to remember those. But in real Schnapsen games against a human opponent, it is very important to remember both players’ trick point totals. I need a lot more practice keeping these totals and the cards all in my head simultaneously.

There is one more thing a good Schnapsen player should remember, which is any card your opponent holds that you have seen. Such a card could be a trump that was exchanged, or a marriage partner from a marriage declared previously. These are usually memorable enough that I find they don’t need to be included in my explicit recitation.

If you have some other memory scheme that you use, or you have questions about mine or my friend’s, please leave a comment below or send me mail. I always enjoy hearing from readers.

© 2013 Martin Tompa. All rights reserved.

Protection

2013-01-21T11:00:00-08:00

The Schnapsen Log

January 21, 2013

Protection

Martin Tompa

It’s a cold winter day with ice on the ground. But you have made your way safely enough to the comforting warmth of the Café Abeles, where Peter waves to you from his usual table near the fire. He already has a generous piece of Haselnusstorte and a steaming cup of coffee with whipped cream in front of him, so you order for yourself while he deals out the cards. The first few deals are uneventful but, before your coffee has cooled enough to drink, you find yourself in the following situation:

Unseen cards:
♠ —
♥ Q
♣ TKJ
♦ AK

Your cards:
♠ K
♥ K
♣ A
♦ TQ

Trump: ♥J
Stock: 1 face-down card
Game points: Peter 5, You 7
Trick points: Peter 31, You 21
On lead: Peter

Peter pauses to consider what to do next, and finally leads ♣T. Plan your play while you sip coffee. Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

The Lowly Jack

2013-01-10T11:00:00-08:00

The Schnapsen Log

January 10, 2013

The Lowly Jack

Martin Tompa

It’s come down to the very last deal in this close game against Polite Polona. Each of you needs only one more game point to win when you find yourself in this position.

Unseen cards:
♠ K
♥ TK
♣ A
♦ QJ

Your cards:
♠ —
♥ A
♣ K
♦ ATK

Trump: ♥J
Stock: 1 face-down card
Game points: Polona 1, You 1
Trick points: Polona 40, You 24
On lead: Polona

With an eerie sense of déjà vu, you notice the remarkable similarity of this position to the deal you just finished playing. This time, however, instead of leading a high trump, Polite Polona chooses to lead the lowly ♦J. Can you come from behind to win this game? Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

Who's in Control?

2013-01-03T11:00:00-08:00

The Schnapsen Log

January 3, 2013

Who’s in Control?

Martin Tompa

Your Sunday family game against Polite Polona has been extremely close, with your Uncle Hans looking on. It’s coming down to the wire. Aunt Polona needs only one more game point to win and you find yourself in this position.

Unseen cards:
♠ Q
♥ TK
♣ A
♦ TQ

Your cards:
♠ —
♥ AQ
♣ K
♦ AK

Trump: ♥J
Stock: 1 face-down card
Game points: Polona 1, You 2
Trick points: Polona 46, You 18
On lead: Polona

Not only is your Aunt Polona close to her goal in game points, but she’s also far ahead in trick points on this deal. She fingers each of her cards in turn and finally chooses to lead ♥T. Can you still save this game? Once you have a good plan, you are welcome to read my analysis.

© 2013 Martin Tompa. All rights reserved.

The Last Trick

2012-12-30T11:00:00-08:00

The Schnapsen Log

December 30, 2012

The Last Trick

Martin Tompa

“I’m curious,” your Uncle Hans says to you. “In the previous deal, you led ♠A from this position.” Hans recreates the position on the table in front of you.

Unseen cards:
♠ TQ
♥ AQ
♣ KQ
♦ —

Your cards:
♠ AK
♥ J
♣ —
♦ AT

Trump: ♦J
Stock: 1 face-down card
Game points: Polona 2, You 2
Trick points: Polona 34, You 12
On lead: You

“When we were discussing your play after the deal, you mentioned that you saw a way to win if the last remaining face-down card in the stock was ♣K. Let’s suppose that was the case, and that your Aunt Polona discarded her ♣Q on your lead of ♠A. This would be the new position after drawing the last cards from the stock.” Hans deftly moves a few cards around on the table.

Polona: (34 points)
♠ TQ
♥ AQ
♣ —
♦ J

You: (26 points)
♠ K
♥ J
♣ K
♦ AT

“It doesn’t seem obvious how you would win from this position,” Hans goes on. “Let’s see, you would be on lead. What was your plan?”

Can you see how to win? Once you have a good plan, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

The Homewrecker

2012-12-28T11:00:00-08:00

The Schnapsen Log

December 28, 2012

The Homewrecker

Martin Tompa

“Ready for the next deal?” your aunt, Polite Polona, asks sweetly. She shoots a quick look at your Uncle Hans, who is kibitzing you, as if to warn him to behave.

Unseen cards:
♠ TQ
♥ AQ
♣ KQ
♦ —

Your cards:
♠ AK
♥ J
♣ —
♦ AT

Trump: ♦J
Stock: 1 face-down card
Game points: Polona 2, You 2
Trick points: Polona 34, You 12
On lead: You

You know that the first thing you should consider, when on lead to the last trick before the stock is exhausted, is closing the stock. You take a quick inventory of your tricks and decide that closing the stock is nearly hopeless. Your current 12 trick points just isn’t enough of a starting platform, even if you could manage to collect 2 diamond and 2 spade tricks.

Thinking about the cards you haven’t yet seen, you take note of the mixed news. The good news is that Polona has no trumps in her hand. The bad news is that she might already be holding the club marriage. In fact, you realize, there is a 2/3 probability that she is, since any of the 4 other cards out of the 6 you haven’t seen could equally well be face-down in the stock. This means that leading ♥J is probably out.

But since your aunt has no trumps, this could be an opportune time to cash ♠A and hope for a good draw from the stock. You count trick points again. Your current 12, plus ♦AT and ♠A, plus at least 8 points contributed by Polona, comes to 52 trick points. If you draw either spade or ♥A, that would give you enough to reach 66. Even if Polona doesn’t hold the club marriage and you draw ♣K, you think you can see a way to take the last trick. That’s good enough odds for you.

You lead your ♠A and Polona discards ♠Q. Unfortunately, you draw the miserable ♥Q from the stock, putting you on lead in this position:

Polona: (34 points)
♠ T
♥ A
♣ KQ
♦ J

You: (26 points)
♠ K
♥ QJ
♣ —
♦ AT

“Oh well,” you sigh, showing everyone your hand. “That ♥Q was an unlucky draw for me. I’ll pull a round of trumps so that I pass 33 points, but then I’m stuck. I can still make one more trump trick after that, but the remaining tricks are yours, Polona. One game point for you.”

“It was a valiant effort, dear,” your aunt tells you, “but it wasn’t meant to be.”

“Or perhaps it was meant to be,” Uncle Hans puts in quietly, unable to keep it to himself.

What is it that Uncle Hans has in mind? Once you have thought about it, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

Too Clever By Half

2012-12-23T11:00:00-08:00

The Schnapsen Log

December 23, 2012

Too Clever By Half

Martin Tompa

Your aunt, Polite Polona, has taken Emmi’s seat in this family gathering. Uncle Hans continues to look over your shoulder and waits to edify.

Unseen cards:
♠ —
♥ TQJ
♣ AK
♦ K

Your cards:
♠ —
♥ AK
♣ T
♦ TQ

Trump: ♦J
Stock: 1 face-down card
Game points: Polona 2, You 4
Trick points: Polona 21, You 25
On lead: You

You list the cards you haven’t yet seen silently to yourself. You see a possibility of an endplay in hearts and try to find your way to it. When you think you may have found it, you close the stock in order to secure 2 game points. It pains you to give up your ♣T to her ♣A, but there’s no way that can be avoided. Well, then, that’s your exit card for the endplay. You lead ♣T, Polona wins the trick, and she exits safely with ♣K.

This looks bad, because you will trump it and pull her last trump with ♦T, but then you will be the one who has to open up the heart suit. You add up the trick points: your current 25, 21 more for the two trump tricks, and 13 or 14 for the ♥A trick. That comes to only 59 or 60, not enough.

And then, suddenly, inspiration strikes you. “Watch this, Hans,” you declare excitedly. Instead of trumping Polona’s ♣K with ♦Q, you instead unblock by trumping unnecessarily high with ♦T. You now have 39 trick points. You then lead your carefully preserved ♦Q as an exit card for the endplay, which Polona wins with ♦K. “If you’ve got the ♥T you’re endplayed, Polona,” you announce, showing her your two remaining hearts. “I have 39 trick points, and these two heart tricks make it at least 66.”

Polite Polona shows you her ♥TJ and, with a gracious smile, says, “That was a beautiful play, dear. How did you ever think of it?”

“It just came to me,” you reply. “Thank you, Polona.”

Uncle Hans clears his throat quietly. Polona turns her head toward him, raises her eyebrow in warning, and says, “Yes?”

Hans quickly turns to you and affirms, “Yes, that was a lovely throw-in, dear. Unblocking the ♦T was very clever. That sort of play is difficult to find, and it worked out beautifully. However, my dears, I just wanted to mention something. Each of you made a mistake in this endgame.”

See if you can spot the mistake Polona made and the one you made. Once you’ve thought it through, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

Postponing Defeat

2012-12-15T11:00:00-08:00

The Schnapsen Log

December 15, 2012

Postponing Defeat

Martin Tompa

We’re back in your house, continuing the epic battle against your sister Emmi. Your clever Uncle Hans looks on expectantly as you consider your next predicament.

Unseen cards:
♠ AT
♥ KQ
♣ J
♦ J

Your cards:
♠ J
♥ J
♣ —
♦ ATQ

Trump: ♦K
Stock: 1 face-down card
Game points: Emmi 2, You 5
Trick points: Emmi 55, You 21
On lead: You

You find yourself fretting about how close Emmi is to 66 trick points and what you’re going to do with those two miserable jacks. Instead of worrying, calm down and plan your play of the rest of this deal. Once you’ve made a plan, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

Setting Up the Endplay

2012-12-06T11:00:00-08:00

The Schnapsen Log

December 6, 2012

Setting Up the Endplay

Martin Tompa

I played a fascinating deal against Doktor Schnaps just a few days ago that I thought you might enjoy working out. This one had a type of endgame that I’d never encountered before.

Doktor Schnaps: (19 points)
♠ AJ
♥ —
♣ J
♦ AQ

You: (26 points)
♠ K
♥ A
♣ AT
♦ T

Trump: ♣
Stock: Exhausted
Game points: Doktor Schnaps 4, You 3
On lead: You

There are only five cards left in each hand, so how difficult can it be to find the winning play? The answer in this case, I think, is that it’s not that easy. But if you figure it out, I think you’ll find it very satisfying. Once you’ve made your plan, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

Make It Progressive

2012-11-30T11:00:00-08:00

The Schnapsen Log

November 30, 2012

Make It Progressive

Martin Tompa

Your losing battle against Emmi continues, but you are expecting a change of luck any deal now.

Emmi: (51 points)
♠ Q
♥ K
♣ Q
♦ AT

You: (40 points)
♠ J
♥ TJ
♣ J
♦ J

Trump: ♥
Stock: Exhausted
Game points: Emmi 2, You 6
On lead: You

Isn’t it just your luck to be behind in trick points, holding a fistful of jacks? Well, make the best you can of the situation. Once you’ve made a plan, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

Putting On Pressure

2012-11-25T11:00:00-08:00

The Schnapsen Log

November 25, 2012

Putting On Pressure

Martin Tompa

It’s another pleasant Sunday afternoon at home, with all the relatives gathered at your house for coffee, cake, and cards. You are playing against your sister Emmi, with your Uncle Hans kibitzing. You hate losing to her, but it seems to be happening again.

Emmi: (33 points)
♠ AT
♥ A
♣ K
♦ Q

You: (40 points)
♠ K
♥ T
♣ TJ
♦ J

Trump: ♣
Stock: Exhausted
Game points: Emmi 3, You 6
On lead: You

Emmi has been lording it over you again and is 3 game points ahead. Things look bleak for this deal as well. You study your hand, review the cards you know Emmi holds in hers, and count up the points you can collect. Once again Emmi has managed to accumulate the high cards in nearly every suit. Her constant good luck drives you crazy!

You throw your hand face up on the table, announcing, “I give up. All I’ve got are my two trump tricks, on which you’ll discard two small cards, and I won’t even get to 60 trick points. The other tricks are all yours. You get another game point.”

“OK,” agrees Emmi quickly, putting her hand face up on the table and recording her game point. “I’ll take it.”

“I don’t understand it, Hans,” you complain. “How is it that Emmi always picks up all the aces and tens?”

“That is bad luck for you, dear,” your uncle commiserates. “But do you think that sometimes your frustration gets in the way of thinking through the situation?”

Is Uncle Hans just offering avuncular life advice, or does he have something more specific in mind? When you think you know the answer, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

More on Race to the End

2012-11-23T11:00:00-08:00

The Schnapsen Log

November 23, 2012

More on Race to the End

Martin Tompa

Florian Wisser and I have been having some intriguing conversations about a column I posted recently entitled Race to the End. (Faithful readers will recognize Florian as the author of the formidable Doktor Schnaps program.) What emerged from those conversations is that there is much more to the deal I presented in that column than I had realized. Here was the position I provided:

Unseen cards:
♠ K
♥ K
♣ AKJ
♦ T

Your cards:
♠ A
♥ A
♣ TQ
♦ J

Trump: ♥J
Stock: 1 face-down card
Game points: Katharina 1, You 3
Trick points: Katharina 21, You 25
On lead: You

In my analysis, I concluded that you should close the stock, play ♥A to pull Katharina’s last trump, play ♠A to pull her last safe exit card, and throw her in by leading ♦J. She would then be forced to open up the club suit, and the trick you would win with your ♣T would bring you to at least 67 trick points. I admitted that this line of play would fail if ♦T was still in the stock, since in that case your throw-in would fail, but that this still gave you a 5/6 probability of winning 2 game points.

What I missed was the possibility of a clever play by Katharina, if either ♥K or ♠K was still in the stock. In either of those cases, Katharina might make an unblocking play by discarding her ♦T! If she does this, you will have 61 trick points, but your throw-in will be foiled. You can cash ♦J, collecting her ♣J and 4 more trick points, but then you are the one who will have to open up the clubs and Katharina will take the remaining two tricks. If Katharina makes this insightful unblocking play, you have only a 3/6 probability of winning the deal, not 5/6, because you will fail if any of ♦T, ♥K, or ♠K is in the stock.

Florian pointed out a clever alternative play that has a better probability of success. You can close the stock, forgo pulling trumps, and lead ♠A, with the intention of next leading ♦J. This line of play will fail if ♠K is in the stock: Katharina will trump your ♠A, cash ♦T and ♣A, and you will end up with only 52 trick points. But the beauty of Florian’s play is that the ♦J will succeed in throwing Katharina in if either ♥K or ♦T is in the stock. In the latter case, you have cleverly left her a trump with which she can be thrown in.

But if, instead, one of those clubs is in the stock, your throw-in will fail. Katharina will win ♦T, but will still have ♥K as an exit card, whose lead will bring you to 55 trick points. Now the outcome will depend on exactly which club is in the stock. If it is ♣J, you will fail, because you are left with two losing clubs. But if it is either ♣K or ♣A, leading your ♣Q next means that you will win the last trick with ♣T, arriving once again at 67 trick points.

The conclusion is that Florian’s line of play only fails if either ♠K or ♣J is in the stock. This gives you a 4/6 probability of winning the deal, better than the 3/6 probability we saw above if you adopt my line of play and Katharina makes the unblocking play when possible.

Having said all this, I would like to go back to my suggested line of play (close the stock and lead ♥A, ♠A, and ♦J) and ask the question, “From Katharina’s point of view, if she is missing either ♠K or ♥K, is discarding ♦T the right play for her to make?” Because remember, she does not know that that missing card is in the stock. From her point of view, it may instead be in your hand.

Let’s suppose first that it is ♠K that is in the stock. In order to analyze the situation from her point of view, we have to stand up, walk around to the other side of the table, and sit in her seat. Here is what the position looks like to her:

Unseen cards:
♠ AK
♥ A
♣ TQ
♦ J

Katharina’s cards:
♠ —
♥ K
♣ AKJ
♦ T

Trump: ♥J
Stock: 1 face-down card
Game points: Katharina 1, You 3
Trick points: Katharina 21, You 25
On lead: You

You close the stock and lead ♥A, on which she plays ♥K, bringing you to 40 trick points. Next you lead ♠A and Katharina has arrived at the critical juncture. Is her best play on this trick to discard ♦T? If you do not hold ♠K, we know that it is not only her best play, but the only play she can make to stop you from winning the deal. But if you do hold ♠K, it is a mistake for her to discard ♦T on your ♠A. That trick will bring you to 61 trick points, and cashing ♠K will then give you enough. If she instead discards ♣KJ on your two spades, you will reach only 61 trick points in total, and she will have to win the last two tricks with ♣A and ♦T whatever your remaining cards are.

So, from Katharina’s point of view, is it more likely that you hold ♠K or that you do not? From a purely probabilistic point of view, with no other information, there is only a 1/4 probability that ♠K is the one card in the stock out of the four Katharina has not seen up to this point.

But this point of view is foolish, because Katharina has other information she should use. In particular, she knows that your specific cards warranted closing the stock. Would you have elected to close the stock holding only the three winners ♥A and ♠AK, knowing that all she has to do to defeat you is to discard three low cards on these winners? This is a difficult question to answer. If Katharina’s answer to this question is no, then she will conclude that ♠K must be in the stock and her correct play is to unblock her ♦T on your ♠A. But if that is how she is certain to play, then her answer to the question should be yes, you would close the stock with only those three winners, since Katharina is certain to give you the gift of ♦T. The logical conclusion of this is that Katharina cannot be certain you wouldn’t close the stock holding ♠K. She is truly in a dilemma.

The situation is a little harder for Katharina if it is ♥K in the stock rather than ♠K, because in that case she has to make her unblocking decision one trick earlier.

This deal turned out to be much more involved and complex than I had originally realized. Many thanks to Florian for pointing this out and pursuing this interesting analysis with me. It’s much more fun doing it together with a reader!

© 2012 Martin Tompa. All rights reserved.

Vienna Tournament

2012-11-16T11:00:00-08:00

The Schnapsen Log

November 16, 2012

Vienna Tournament

Martin Tompa

I am going to break from our normal format for today’s column, in order to give you a brief travel report.

I was in the beautiful city of Vienna not too long ago for a visit. Normally the Schnapsen tournament season is during the cold months, and it was still summer weather at the time of my visit. But by good fortune, at the end of the week there was a small, off-season Schnapsen tournament in the outlying district of Hietzing, and I was able to enter.

The weather was so gorgeous that most of the tournament was played outdoors. The tournament mechanics were interesting. Because there is a fair amount of luck involved in the cards you pick up, you can buy up to 4 entries to the tournament. Each entry gives you a ticket for a single round 1 game (just the usual 7 game points) against an opponent. For each round 1 game that you win, you receive a ticket for a single round 2 game. When no one has any round 1 tickets remaining, round 2 begins. This pattern of rounds continues until you have no ticket left, at which point you are knocked out of the tournament. The last player remaining in the tournament is the winner.

I wish I could report that I played skillfully and got further in the tournament than I did. The truth, though, is that I didn’t play very well and got knocked out in round 2. Entering and playing in the tournament was a good experience, and it was fun to meet and play with others who are serious about Schnapsen. I think the next time I’m fortunate enough to be in Austria for a tournament, I will know better how to prepare for it. I hope to get that chance one day. If you are looking for Austrian tournaments, you can find schedules of upcoming events in some of my sidebar “Links in German”.

The other Schnapsen-related part of this trip is that I spent a very pleasant afternoon and evening with Florian Wisser, the author of the devilish Doktor Schnaps program that I discussed in an earlier column. We spent much of the afternoon discussing Schnapsen programs and some interesting problems related to Schnapsen. In the evening, Florian had very thoughtfully arranged a fun Schnapsen mini-tournament with three of his friends and fellow Doktor Schnaps players at the Cafe Brendl. It was wonderful to meet all of them. I am grateful to Florian for his gracious hosting.

© 2012 Martin Tompa. All rights reserved.

Entries from Thin Air

2012-11-09T11:00:00-08:00

The Schnapsen Log

November 9, 2012

Entries from Thin Air

Martin Tompa

It’s come down to the wire at the weekly Black Eagle tournament. You and the reigning champion Katharina are down to the last deal, with the game point score tied 1-1. The winner of this deal wins the tournament. Onlookers who have been knocked out of the tournament have gathered around your table. You suddenly feel cold and clammy, and you’ve got pins and needles in the tips of your fingers. Yet Katharina looks as cool as a cucumber that’s wearing a funny hat.

Katharina: (27 points)
♠ A
♥ KJ
♣ A
♦ K

You: (29 points)
♠ K
♥ ATQ
♣ K
♦ —

Trump: ♠
Stock: Exhausted
Game points: Katharina 1, You 1
On lead: You

You’ve definitely held your own against Katharina tonight, and things couldn’t be tighter. Not only is the game point score tied, but the trick point score is nearly tied too and there are only five tricks left in tonight’s tournament. When you think you have come up with a good plan, you are welcome to read my analysis.

© 2012 Martin Tompa. All rights reserved.

The Schnapsen Log

Homework on Expected Values

The Schnapsen Log

Homework on Expected Values

Martin Tompa

Rags and Riches

The Schnapsen Log

Rags and Riches

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How Much Do You Trust to Chance?

The Schnapsen Log

How Much Do You Trust to Chance?

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A Multitude of Options

The Schnapsen Log

A Multitude of Options

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To the Brave Go the Spoils

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To the Brave Go the Spoils

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Where's the Endplay?

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Where&#8217;s the Endplay?

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An Unusual Capture

The Schnapsen Log

An Unusual Capture

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On a Wing and a Prayer

The Schnapsen Log

On a Wing and a Prayer

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A Bird in Hand

The Schnapsen Log

A Bird in Hand

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Imperfect Information

The Schnapsen Log

Imperfect Information

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Danger at Every Turn

The Schnapsen Log

Danger at Every Turn

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Schemes for Remembering Cards

The Schnapsen Log

Schemes for Remembering Cards in Schnapsen

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Protection

The Schnapsen Log

Protection

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The Lowly Jack

The Schnapsen Log

The Lowly Jack

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Who's in Control?

The Schnapsen Log

Who&#8217;s in Control?

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The Last Trick

The Schnapsen Log

The Last Trick

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The Homewrecker

The Schnapsen Log

The Homewrecker

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Too Clever By Half

The Schnapsen Log

Too Clever By Half

Martin Tompa

Postponing Defeat

The Schnapsen Log

Postponing Defeat

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Setting Up the Endplay

The Schnapsen Log

Setting Up the Endplay

Where’s the Endplay?

Who’s in Control?