In this final post, I am going to recap the discussed concepts in as few words as possible.

If you want to win at Schnapsen, you need to understand that you have to take as **many tricks containing cards of high value** as possible. Therefore, you need to start taking tricks at some point during a game. **It makes no sense to stay passive**. To reach the 66 card point mark, you must take at least 4 tricks if you do not announce a marriage.

An important concept involves **protecting the lone tens**, which will allow you to take tricks containing those tens. It would be a costly error to carelessly give these points away.

The **strength of your starting hand** makes a big impact on your strategy: If your hand is bad, try to limit the game points your opponent can score. If you have a good hand, try to score big.

If you have a hand at some point during a game that will allow you to score enough points most of the time, **closing the deck** usually is the best strategy. The only exception to this is when your opponent is very unlikely to take the next trick and when you have a significant chance of improving your hand with the next card.

Towards the end of a game, you must take the **dead suits** into account. During a game, you should try to put yourself into a position in which you can take advantage of the dead suits.

The farther you fall behind your opponent during a Bummerl, the **more aggressively** you should play. However, if you are the one that pulls away, try to choose strategies that will not let your opponent get back into the game.

If you try to incorporate the concepts we have discussed into your game, it will improve significantly. I am afraid that the articles may have been very mathematical for the most part, but this is what the difference between winning and losing comes down to. Obviously, we have not covered all possible situations, but you should be able to make the right conclusions in any new situations as you are familiar with the thought process that leads to making the right decisions.

## 2010-10-08

### Final Thoughts

## 2010-10-07

### Probabilities

In this post, I am going to provide the most important probabilities in Schnapsen. Please note that the following tables are correct only as long as players receive additional cards from the deck after each trick. So if the deck is closed, these probabilities do not apply.

The following table displays the **probabilities at the start of the game**:

This table tells you the probabilities of your opponent holding x trumps cards (P(x)) given that you have a certain number of trump cards in your hand (this number is listed in the second row).*Example #1*. If you have 2 trump cards in your starting hand, your opponent is 39.6% to have no trump cards at all.

While the primary purpose of these tables is to show the probabilities associated with trump cards, you can derive many other probabilities.*Example #2*. A certain strategy option you have only works if your opponent does **not** have one specific card. How likely is this scenario? Basically, this situation is the same as the following one: You have 3 trump cards and your opponent has the sole remaining trump card. Therefore, the probability of your opponent having the card is 35.7%. This strategy option will work 64.3% of the time.*Example #3*. A certain strategy option only works if your opponent does **not** have 2 specific cards at the same time (if he only has one of them, you will still succeed). Mathematically, this situation is equivalent to you having 2 trump cards and your opponent not having the remaining 2 trump cards. Therefore, this strategy option is going to work 89.0% of the time.*Example #4*. You would prefer your opponent not to have 3 specific cards. This is equivalent to you having one trump card and your opponent not having any other trump card. So, the associated probability is 23.1%.*Example #5*. This example might not be relevant in an actual playing situation, but it helps you understand how to use the probability tables: The 2 cards of interest are the Ad and the Tc. What is the probability of your opponent holding the Tc while not having the Ad? Your opponent is going to have the Tc 35.7% of the time. Furthermore, he will have both the Tc and the Ad 11.0% of the time. Consequently, your opponent is 24.7% to hold the Tc without having the Ad.

During a game, these probabilities change. **After the first trick**, the probability table looks like this, given that **all trump cards are still live**:

Now, let's say that **one trump card has left the game**. Then, the table would look like this:

As you can see, the probabilities are the same, but have moved one column to the left. So, it is rather easy to determine the tables if trump cards are out of the game: You move the figures to the left by as many columns as trump cards are out of the game.**After the second trick**, the table looks like this:

Please note that your opponent is exactly 50% to have any one specific card at this point.

The probabilities **after the third trick** are the following ones:

Finally, **after the fourth trick**, the table looks like this:

While it might not be necessary to exactly know all of these figures, you should be able to roughly estimate how (un)likely certain scenarios are. This will help you estimate the winning percentages of strategy decisions more accurately.

## 2010-10-06

### The Tough Rules

In all previous posts, we have assumed that the game is played according to the basic set of rules. Then, we have identified sensible strategies. In Schnapsen tournaments however, there are additional rules that make the game a bit more difficult to play. In this article, I will outline how these additional rules affect your game and your strategy.*Additional Rule #1: Tricks are taken by the winner, immediately turned face-down, and must not be looked at again.*

There is no need for you to change your strategy because of this rule. You simply have to keep track of the game, count the cards that have left the game, and count the card points. As I have stated before, you should do all of these things anyway.*Additional Rule #2: The turned-up trump card cannot be exchanged for the trump jack right at the start of a game or if the stock is down to one card.*

This rule has some implications: At the start of a game, the player who has the lead usually has an advantage. This rule basically takes away this advantage, especially in combination with additional rule #5. Still, this does not affect your strategy in any way as there is nothing you can do about it.

If you get the Jt at some point during a game, this rule will not have much of an impact on your game: If you want to win, you need to start taking tricks at some point, so most of the time you will get a chance to exchange the Jt for the other trump card. However, you should expect the face-up trump card to be a card of higher value more often than under standard rules. Consequently, you will more often find yourself in a position in which you have to make a difficult decision: Would you rather take the face-up trump card or have the lead once the deck is gone? Obviously, this decision depends on the actual situation, but there are a number of interesting scenarios that could occur: The last trump card could give your opponent the trump marriage; it could give you the trump marriage; the face-up trump card could be the At. It must be stated that just holding the trump marriage is not enough. You have to actually announce it to score 40 points. So, if you hold the trump marriage, but your opponent has the lead, you will often be forced to trump and thereby lose your marriage. Furthermore, if additional rule #6 applies, it does not matter if anybody has a marriage after the deck is gone.*Additional Rule #3: The deck may not be closed if it is down to one card.*

Usually, once the deck is down to one card, the leading player has a very good idea about the opponent's cards: Of the 6 live cards she has 5. Therefore, it is very easy to estimate the success rate of closing the deck, while the opponent can do nothing about it. This rule takes away the huge edge the leading player has. If you consider closing the deck, you have to take trick #3, as the deck may not be closed after trick #4. There is more uncertainty involved, yet good players should not have a hard time coping with this situation.*Additional Rule #4: After announcing a marriage, the king has to be played.*

The only difference between playing the king and the queen is a single point in card value. Obviously, in certain situations, this single point might make a difference, but usually it does not really matter.*Additional Rule #5: Marriages may be announced only if a trick has already been taken.*

Just like additional rule #2, this rule takes away one of the advantages the starting player has. Only the starting player is affected by this rule: If you want to announce a marriage at any other time, this implies that you have already taken at least one trick. Now, most of the time it is pretty clear which card to lead with: a low card that is not part of the marriage. The most interesting situation occurs when you have the marriage and 3 cards of high value: If you have the highest trump card, you might decide to play that card and then announce the marriage, which will put you above the 33 card point mark in most cases. This is not a bad option, especially if you have unprotected tens in your hand. If you cannot be sure to take the first trick, however, you should consider giving up the marriage: Let's look at an example. You decide to play a ten. Your opponent is 36% to have the ace of that suit, in which case she would score 21 points with that trick. Furthermore, depending on the actual situation, your opponent might be more or less likely to have trump cards. In both cases, if your opponent takes the trick, you might end up not announcing the marriage at all.*Additional Rule #6: Marriages may not be announced once the deck is gone.*

If you hope to catch a marriage yourself, you have to try to be in charge during the fifth trick at the latest, as after trick #5 no marriages may be announced. You can use this rule to your advantage, as well: If there are live marriages that your opponent might have, you can keep him from announcing them by taking the fourth trick.

## 2010-10-03

### Dead Suits and the Final Tricks of a Game

This article is going to deal with two separate topics: the impact of dead suits on your strategy and how to approach the final 5 tricks of a game.**Dead suits**

Usually, the cards worth few points get played early on in a game, and at the end both players are armed with cards of high value and trump cards. However, in general this is not a sensible strategy; a few adaptations should be made: Aces are not vulnerable in any way; therefore, you can wait until your opponent plays a card of that suit and then take the trick. The situation with tens is different, as long as the aces of the corresponding suits are still live.

Let's say you have the Qs and the Ts, all other spades are still live. In this situation, I would not recommend playing the Qs: First of all, you still have the chance to catch the Ks, which would give you a marriage. And second, if you got into a situation in which suit had to be followed, you would like to have a spade of low value in your hand, in case your opponent played the As.

If you had the Qs and the Ts, while your opponent had the Ks and the As, you would give your opponent the chance to maximize the card points he could score if you played the Qs: He would likely take the trick by playing the Ks, figuring that he still had the chance to take another trick (potentially of high value) with the ace.

So in a nutshell, one should avoid holding lone tens.**Holding 3 Cards of one Suit**

Another critical situation arises, when you hold 3 cards of a single suit at some point during a game: In this case, you know for sure that towards the end you will hold at least one card of a dead suit. Depending on the trump cards that you will hold and that will still be live at the end, that might be good or bad news. If you have the edge as far as trumps are concerned, you do not care about cards belonging to dead suits, but if you do not, your opponent has the chance to take tricks by trumping and still retain his edge.

Let's look at some examples: You hold the **Js-Qs-Ks**. There is not much you can do, just announce the marriage!

If you hold the **Js-As-Ts**, an interesting situation arises: Please note that your opponent might have the marriage of spades. Ultimately, you would like to take two tricks with the As and the Ts. At the start of a game, you cannot expect your opponent to have the marriage. If you played the Js and your opponent had one of the 2 remaining spades, he would likely decide to take the trick. That would leave you with both the As and the Ts, giving you the chance of taking just one of the spade tricks. In all likelihood, your opponent would take a trick containing a high value spade by trumping at some point during the game. So at the start of a game, you should avoid leading with a spade.

If you hold those 3 cards at the end of a game and all other spades are still live, you might want to decide to play the Js: Your opponent would likely have the marriage and might not want to give it up. Please note that this would not make any sense: If your opponent held onto the marriage, he still would not get a chance to announce it as you would play your spades during the next tricks when the deck is gone. So obviously, there are no upsides to playing the Js at any point. Still, a lot depends on the other cards in your hand and the live trump cards, so you might find yourself in a situation that forces you to play the Js, anyway. Then again, combined with the At-Tt, the As-Ts-Js would be a monster hand.

In the final example, you hold the **Js-Qs-Ts**. This is a very problematic combination of cards. Here, you would like to take one specific trick: Ts-Ks. If you are to lead, you basically cannot play any spade: If you played the Js or the Qs and your opponent had the Ks, he would likely take the trick with that card, which would make it very difficult for you to take a trick with the Ts. If you decided to lead with the Ts, your opponent would decide to trump (if he had any trump cards). Still, at the start of the game, when your opponent is just 36% to have the king, playing the Js might be a viable option.

It should be mentioned that in the examples mentioned above, we just took a look at the cards of a single suit isolated from the rest of the hand. In an actual playing situation, you have to take the situation of trump cards and the other cards in your hand into account, as well, before making any decisions.**The Final 5 Tricks**

When writing about the final 5 tricks, I specifically mean the situation after the deck is gone. First of all, if you have paid attention to the first 5 tricks, you know exactly which cards your opponent has left. The next thing to figure out is how to reach 66 card points as fast as possible without your opponent reaching that mark first. Of course, sometimes you might be in a situation in which you are just happy to get to more than 33 points, but that is not the norm. You must try to take advantage of the situation of trump cards and of dead suits.

You can do that in the following way: If you have cards of a dead suit, by playing them you force your opponent to get rid of a trump card as he has to trump. This might be helpful if your opponent has more trumps than you do. However, please note that if he has cards of dead suits as well, he can do the same to you.

There is no more general advice one can give. Basically, you have to identify the sequence of cards that let you win the game. As in the case of a closed deck, the constraints put upon players by the rules limit the available strategies.

## 2010-10-02

### Closing the Deck

By looking at all the previous posts, it becomes obvious that closing the deck is an important part of Schnapsen. It might even be a player's single most important ability to recognize when to close the deck and to be able to put oneself in a position to actually close the deck at that point. In this article, we are going to have a closer look at the dynamics of closing the deck and at how to successfully adapt one's strategy towards the end of a Bummerl. It might not be apparent at first, but these topics are related to each other.**Consequences of Closing the Deck**

First of all, it must be stated that after closing the deck, the rules change. Players must follow suit or trump depending on the actual situation. Consequently, the player who does not have the lead is constrained as far as strategy is concerned. Because of these limited options available to the opponent, the player who decided to close the deck can more accurately estimate what might happen during the remainder of the game. It is easier to estimate the outcome. Therefore, you should always close the deck as soon as you figure that you have a *good* chance to reach 66 card points. What *good* actually means often depends on the specific circumstances.

If you play against a weak opponent who makes lots of mistakes, closing the deck might not always be the best strategy, as the constraints put upon players by closing the deck take away this player's chance to commit any major errors. Of course, if closing the deck is the best strategy against a good player in a particular situation, it cannot be a bad option against a bad player in the same situation, but it just might not be the best strategy.**Before Closing the Deck**

Whether you should actually close the deck in a particular situation – and this should not come as a surprise to you - depends on the cards in your hand. It is your goal to reach 66 card points. Before closing the deck, you have to estimate the number of tricks you are likely to take and how many additional points you can score via the cards your opponent surrenders. Usually, you have to have the highest remaining cards of live suits and a couple of trump cards. Obviously, it is difficult to give general advice as the cards that are out of a game and the points you have already scored play an important role.

However, the one thing you always have to consider before closing the deck is your winning percentage and the associated EV. As the game points that can be scored depend on the players' card points at the time the deck is closed, it is absolutely clear what is at stake. In a regular game, the possible payoffs are the following ones:

-) either you: +3, or opponent: +3 (blue line);

-) either you: +2, or opponent: +2 (red line);

-) either you: +1, or opponent: +2 (green line).

Obviously, if the game points you and your opponent can score are the same, a winning percentage above 50% is associated with a positive EV. However, if you can just score a single point, your opponent has the chance to score 2 points if you fail to reach 66 card points. Therefore, you need a winning percentage of 67% to break even.

At the beginning of a game, you should only close the deck if this decision yields a positive EV. If that decision has a negative EV, it is usually better to wait and see how the game develops. Towards the end of a game, you might want to close the deck even if that decision has a negative EV: This situation occurs if the decision to continue without closing the deck would have a negative EV and the option to close the deck, while still having a negative EV, has a slightly higher EV.**Towards the End of a Bummerl**

Once a player needs less than 3 game points to win at the end of a Bummerl, the payoff structures change. Let's look at an example. Your opponent is ahead, the score is 1-4. In this situation, the highest number of game points she can score in the next game is a single point, while you can still score anywhere from 1 to 3 game points. Now the diagram depicting the relationship between winning percentage and EV looks different from the one above:

In a nutshell, you should play more aggressively and try to score big if you have some chance to do so. If you can score 3 game points, you need a winning percentage of just 25% to break even. For 2 game points, that figure is 33%. As you are far behind, it makes no sense to settle for a low score. You should take the risk of losing and thereby give yourself a better chance to win the Bummerl in the following game.

If your opponent is 2 game points away from a win, the situation is similar:

However, you need slightly higher winning percentages to yield a positive EV.

These numbers should not only influence your strategy when you are behind at the end of a Bummerl, but when you are ahead as well. There is no need to try and score 3 game points, when you can actually score but a single point. By taking unnecessary risks, you give your opponent the chance to get back into the game.