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Counting card points in Tarot games

[from http://www.netlink.co.uk/users/pagat/tarot/counting.html ]

Subject: Counting card points in Tarot games
                                   
     * Introduction
     * Which cards are worth points
     * Grouping the cards
     * The original method: adding a point for each trick (or group)
     * Second method: fractional values
     * Third method: counting by subtraction
     * Fourth method: counting the valuable cards
     * Incomplete groups
     * Comparison of the methods
     * Counting the Fool in classic Tarot games
       
Introduction

   This page is relevant to most games played with Tarot cards.
   These names of these games vary slightly according to the
   language of the place where they are played - for example Tarot,
   Tarok, Tarocchi, Taroky, Tarock, Tarokk, Troccas, Trogga and
   Droggn. There are also Tarot games known additionally or instead
   by other names such as Cego, Königrufen, Zwanzigerrufen,
   Paskievics, Ottocento and many others.
   
   These are all trick taking games in which the cards have point
   values. At the end of the play, each player or team will have a
   pile of cards they have taken in tricks (or acquired in some
   other way such as from the talon or by discarding them). They
   then count those cards to decide whether they have won or lost.
   
   This page is purely about how to count the cards. The method is
   essentially the same for nearly all Tarot games. As it can be
   somewhat puzzling to newcomers to these games, it is described
   here fully to save repetition of the full explanation on the page
   for every such game.
   
Which cards are worth points?

   The pack generally consists of
     * four suits each containing four picture cards and some
       numeral cards,
     * 21 numbered trumps,
     * the fool, an unnumbered card known by various names such as:
       excuse, sküs, stieß, gstieß, narr, matto.
       
   The valuable cards are the picture cards in the suits, the 1 and
   21 of trumps and the fool. The remaining cards - the 2 to 20 of
   trumps and the numeral cards in the four suits - are hereafter
   called empty cards.
   
   The values of the cards as usually quoted and most familiar to
   Tarot players will be called the nominal values. For the purpose
   of some of the alternative counting methods described below,
   other values need to be used, which I call the reduced values,
   and the fractional values. The values under each of the three
   systems are set out in the following table (in which n represents
   the size of the groups in which the cards are counted in the
   game):
   
   cards nominal value reduced value fractional value
   1, 21 and fool 5 points 4 points 4 + 1/n points
   kings 5 points 4 points 4 + 1/n points
   queens 4 points 3 points 3 + 1/n points
   riders 3 points 2 points 2 + 1/n points
   jacks 2 points 1 point 1 + 1/n points
   empty cards 1 point nothing 1/n point
   
    Exceptions
     * In some Italian games, there are additional trumps that are
       worth 5 points, and in some cases, certain trumps are worth
       10 points.
     * In the Austrian game Zwanzigerrufen the empty trumps have a
       nominal value of 1 point as usual, but the numeral cards in
       the suits are worth nothing at all - their nominal and
       fractional value is zero and their reduced value is
       effectively minus 1.
       
Grouping the cards

   The complexity in the counting comes from the fact that in most
   games, the cards are counted in groups. Most commonly they are
   counted in groups of three cards; in some games the cards are
   counted in groups of two or four; and in some they are in fact
   counted in groups of one (individually).
   
   When counting in groups, it does not matter how the cards are
   arranged into groups. When counting in threes, take any three of
   your cards, then any other three and so on. Some people count
   them in the order they lie in the pile, but it can be easier to
   rearrange them to make the addition easier. Mixing the cards up
   and rearranging them into new groups never changes the total
   value of the pile.
   
The original method: adding a point for each trick (or group)

   This is the easiest method to understand, and was certainly the
   original method. However it is not the fastest, so you will
   probably prefer to use one of the other methods once the
   principle is understood.
   
   For this method, the reduced values of the cards are used. You
   simply add up the values of all your cards and then add one extra
   point for each group. (Incomplete groups are discussed below.)
   
   Using this method it clearly does not matter which cards you put
   into which groups - the number of groups and the total of the
   card values remain the same.
   
   It seems certain that in early forms of Tarot the groups
   originated as tricks, and the size of the groups was therefore
   equal to the number of players in the game. So the original
   method of counting was to count the reduced value for each
   valuable card plus one extra point for each trick taken. Many
   modern games have lost sight of the relationship between groups
   and tricks, so it is common to find, for example, a four player
   game in which the cards are counted in groups of three.
   
   Examples (counting in threes):
     * a group of three empty cards is worth one point (nothing for
       the cards and one for the group)
     * a king plus two empty cards is worth 5 points (4 for the king
       and one for the group)
     * a king, a queen and an empty card are worth 8 points (4+3+1)
     * three riders are worth 7 points (2+2+2+1)
       
   Note that in the extreme case of a game where the cards are
   counted in ones every card is a group, so you would just add one
   point to the value of every card. Rather than doing this it is
   quicker and easier just to use the nominal values.
   
Second method: using fractional values

   In this method, instead of adding one point for each group we add
   a fraction of a point to the value of each card, to give the same
   result. The total value of a pile of cards is simply the sum of
   the fractional values of the cards it contains.
   
   This method is often found in descriptions of French Tarot, where
   n=2, so the fractions are halves.
   
   Examples (counting in threes):
     * a group of three empty cards is worth one point (one thrid of
       a point for each card)
     * a king plus two empty cards is worth 5 points (4 1/3 + 1/3 +
       1/3)
     * a king, a queen and an empty card are worth 8 points (4 1/3 +
       3 1/3 + 1/3)
     * three riders are worth 7 points (two and one third points
       each)
       
Third method: counting by subtraction

   In this method the nominal values of the cards are used, but if
   the cards are counted in groups of more than one, adding the
   nominal values gives a total value for each group which is too
   high. If the cards are counted in groups of n, you have to
   subtract n-1 points from the total value of each group. For
   example, when counting in threes, you add up the values of the
   cards in a group and subtract two.
   
   Examples (counting in threes):
     * a group of three empty cards is worth one point (1 + 1 + 1 -
       2)
     * a king plus two empty cards is worth 5 points (5 + 1 + 1 - 2)
     * a king, a queen and an empty card are worth 8 points (5 + 4 +
       1 - 2)
     * three riders are worth 7 points (3 + 3 + 3 - 2)
       
   This method may sound unnecessarily complicated, but in fact it
   is the most practical of the methods so far discussed, and is
   quite widely used. Notice that in the common case of a group
   consisting of a single valuable card and some empty cards, the
   total value of the group is just the nominal value of the
   valuable card, because the 1 point for each of the n-1 empty
   cards is cancelled by the n-1 points that you have to subtract.
   
Fourth method: counting the valuable cards

   The nominal values of the cards are used, and the value of a
   group is worked out as follows:
     * A group consisting entirely of empty cards is worth one
       point.
     * A group containing one valuable card is worth the value of
       that card.
     * A group containing two valuable cards is worth one less than
       the sum of their values.
     * A group containing three valuable cards is worth two less
       than the sum of their values.
     * A group containing four valuable cards is worth three less
       than the sum of their values.
       
   This is the most practical method, and the one that most players
   use to count quickly, but it can appear quite mysterious at first
   sight. In particular it might not be immediately obvious using
   this method that the value of your pile of cards stays the same
   no matter how you divide it into groups. This becomes clearer if
   you appreciate that it is really just a streamlined version of
   method three (subtraction).
   
   Examples (counting in threes):
     * a group of three empty cards is worth one point (by
       definition)
     * a king plus two empty cards is worth 5 points (the value of
       the king)
     * a king, a queen and an empty card are worth 8 points (5 + 4 -
       1)
     * three riders are worth 7 points (3 + 3 + 3 - 2)
       
   In practice, when using this method, it helps to rearrange the
   cards you are counting so that as far as possible there is just
   one valuable card in each group.
   
Incomplete groups

   What happens when the pile of cards you are counting does not
   divide exactly into groups? After you have counted as many groups
   as you can, what happens to any odd cards that are left over?
   
   The answer to this depends on the rules of the particular game
   you are playing, but there are some general principles.
   
   When the cards are counted singly, obviously there is no problem.
   
   When cards are counted in twos, the incomplete group will be a
   single card, and using the fractional method, the total will
   contain an odd half point. Generally the rules cause this to be
   rounded one way or the other - for example in French Tarot played
   by an odd number of players, half points are rounded in favour of
   the winning side.
   
   When cards are counted in threes, an incomplete group can contain
   one or two cards, and is dealt with as follows:
   two odd cards one odd card
   original method add one point, as though it were a complete group
   add nothing for the group
   fractional method round two thirds up to one round one third down
   to nothing
   subtraction method subtract one point from the sum of the values
   subtract one point from the value
   counting valuable cards count as though an empty card is added to
   complete the group an empty card is worth nothing
   a valuable card is worth one less than its nominal value
   
   There are not many games in which the cards are counted in fours.
   An example is the Swiss game Troccas, in which the dealing side
   has a pair of cards left over. These are counted as a whole group
   - i.e. as though two extra empty cards were added to them.
   
Comparison of the methods

   It should by now be clear that:
    1. each method produces the same result (under each method, the
       examples show how to calculate the value of the same groups
       of three cards, demonstrating that the results are the same);
    2. it does not matter how the cards in a pile are arranged into
       groups (this is obvious for the first two methods, and must
       therefore also be true for the others, as they give the same
       results);
    3. if the complete pack is divided into two parts (as it
       normally will be at the end of the play between two teams),
       the totals of the two parts always add up to the same total
       value of the cards in the pack.
       
   The original method is straightforward, but suffers from the
   practical drawback that you effectively have to go through the
   cards twice to count them.
   
   The fractional method is also easy to understand in theory. Some
   people find it the best method for keeping track of the value of
   cards taken by each side during the play. For counting your pile
   of cards at the end, it is rather fiddly to operate unless you
   are good at adding up fractions.
   
   The subtraction method has the advantage that it uses the nominal
   values of the cards, which most players recognise, so you will
   not be talking at cross purposes when discussing the game. It is
   also quite practical, and the formula is easiest to remember.
   
   The fourth method, counting valuable cards, is the fastest and
   easiest to use in practice, provided that you can remember it.
   
   Note that when the cards are counted singly, there ceases to be
   any point in the original method, and the other three methods all
   come to the same thing. A useful practical technique for fast
   counting when counting the cards singly is to arrange the cards
   in groups that add up to five - for example a king, a rider and a
   jack, a jack and three empty cards, etc.
   
Counting the fool in classic Tarot games

   By classic games I mean ones in which the fool is not the highest
   trump, but is a card which is played to excuse the holder from
   following suit to the trick. In this case the fool is not taken
   by the winner of the trick to which it was played, but is added
   to the trick pile of the person who played the fool.
   
   The result of this procedure is that the team which took the
   trick to which the fool was played may be one card short, and the
   side which played the fool may have one card too many. There are
   essentially three possible ways of dealing with this:
     Giving a card in exchange for the fool
       The side which have the fool give in exchange an empty card
       from their tricks to the side which won the trick to which
       the fool was played. The counting is then normal.
     Adjusting the count
       The cards are counted as though an empty card was given in
       exchange for the fool, without physically giving the card.
       The side with an extra card count their cards as though they
       had one fewer empty card; the other side count as though they
       had an additional empty card.
     Counting the fool as 4 points alone
       Another way of adjusting the count is that the side with the
       fool counts this card as worth 4 points on its own (not 5),
       and the fool is not included in any group. The side which
       took the trick to which the fool was played count as though
       they had an extra empty card.
       
   All of these three methods give the same result.
     ____________________________________________________________
   
   Return to General Index of Card Games site
     ____________________________________________________________
   
   
    This page is maintained by John McLeod
    (john@pagat.demon.co.uk).
    Last updated 8th July 1997

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