When this page was first written (in late 1995), there was already a fair amount of Bridge-related information on the Internet, but most of it was aimed at people who already knew how to play. This page was produced to fill the gap by explaining how Bridge is played. The explanation is intended for people who have some experience of cards and card games, but no knowledge of Bridge.
In the following years, several other sites with information of use to Bridge beginners appeared. Some of these appear among the links at the end of this page.
- Types of Bridge
- Rubber Bridge
- Duplicate Bridge
- The Beer Card
- Honeymoon Bridge
- Bridge Resources on the Internet
Types of Bridge
Contract Bridge was invented in the 1920's and in the following decades it was popularised especially in the USA by Ely Culbertson. Bridge currently occupies a position of great prestige, and is more comprehensively organised than any other card game. There are clubs, tournaments and championships throughout the world.
Rubber Bridge is the basic form of Contract Bridge, played by four players. Informal social Bridge games are often played this way, and Rubber Bridge is also played in clubs for money.
Duplicate Bridge is the game normally played in clubs, tournaments and matches. The game is basically the same but the luck element is reduced by having the same deals replayed by different sets of players. At least eight players are required for this. There are some significant differences in the scoring. Two types of Duplicate Bridge will be covered:
Chicago is a version of Bridge played by four people over four deals.
Contract Bridge developed in the 1920's from Auction Bridge, which is different mainly in the scoring. In Auction Bridge, overtricks count towards making game, so it is only necessary to bid high enough to win the contract - there is no incentive to bid all the tricks you can make.
Before Auction Bridge there was Bridge-Whist or Straight Bridge (at the time this game was just called Bridge). Here is a link to the earliest published rules of Bridge, which appeared in 1886 under the name Biritch or Russian Whist. In Bridge-Whist there is no bidding at all - the dealer either names a trump suit or passes, in which case the dealer's partner must choose trumps. In either case the dealer's partner is dummy. Either opponent may double before the lead to the first trick, and if doubled, the dealer's side may redouble. In the earliest form of the game, after any redouble, the other side can redouble again, and this can continue indefinitely.
The duplicate format, in which the same cards are played at more than one table, has been in use since the 19th century for competitions in Auction Bridge, Straight Bridge, their ancestor Whist, and several other four-player card games, as well as for Contract Bridge from its invention to the present day.
Players and Cards
There are four players in two fixed partnerships. Partners sit facing each other. It is traditional to refer to the players according to their position at the table as North, East, South and West, so North and South are partners playing against East and West. The game is played clockwise.
A standard 52 card pack is used. The cards in each suit rank from highest to lowest: A K Q J 10 9 8 7 6 5 4 3 2.
The cards are shuffled by the player to dealer's left and cut by the player to dealer's right. The dealer deals out all the cards one at a time so that each player has 13. Turn to deal rotates clockwise.
It is traditional to use two packs of cards. During each deal, the dealer's partner shuffles the other pack and places it to the right. The dealer for the next hand then simply needs to pick up the cards from the left and pass them across to the right to be cut. Provided all the players understand and operate it, this procedure saves time and helps to remember whose turn it is to deal, as the spare pack of cards is always to the left of the next dealer.
There is next an auction to decide who will be the declarer. A bid specifies a number of tricks and a trump suit (or that there will be no trumps). The side which bids highest will try to win at least that number of tricks bid, with the specified suit as trumps.
When bidding, the number which is said actually represents the number of tricks in excess of six which the partnership undertakes to win. For example a bid of "two hearts" represents a contract to win at least 8 tricks (8 = 6 + 2) with hearts as trumps.
For the purpose of bidding the possible trump suits rank as follows: no trumps (highest), spades, hearts, diamonds, clubs (lowest). A bid of a larger number of tricks always beats a bid of a smaller number, and if the number of tricks bid are equal, the higher suit beats the lower. The lowest bid allowed is "one club" (to win at least 7 tricks with clubs as trumps), and the highest is "seven no trumps" (to win all 13 tricks without trumps). NB. In North America, the term for contracts played without a trump suit is "notrump" or "no trump" (without an 's').
It is also possible, during the auction, to "double" a bid by the other side or to "redouble" the opponents' double. Doubling and redoubling essentially increase the score for the bid contract if won and the penalties if lost. If someone then bids higher, any previous doubles and redoubles are cancelled.
Note that doubling does not affect the ranking of a bid - for example a bid of two spades is always higher than two hearts, even if the two hearts bid has been doubled or redoubled.
The dealer begins the auction, and the turn to speak passes clockwise. At each turn a player may either:
- make a bid, which must be higher than the previous bid if any;
- say "double", if the previous bid was by an opponent, and has not already been doubled;
- say "redouble", if the previous bid was by one's own side and has been doubled by an opponent, but not yet redoubled;
- pass, by saying "no bid" or "pass". This indicates that the player does not wish to bid, double or redouble at that turn, but a player who has passed is still allowed to bid, double or redouble at a later turn. NB. Either "no bid" or "pass" is permissible, but you should stick to one term or the other. "No bid" is usual in Britain; "pass" is usual in the USA.
If all four players pass on their first turn to speak the hand is said to be passed out. The cards are thrown in and the next dealer deals.
If anyone bids, then the auction continues until there are three passes in succession, and then stops. After three consecutive passes, the last bid becomes the contract. The team who made the final bid will now try to make the contract. The first player of this team who mentioned the denomination (suit or no trumps) of the contract becomes the declarer. The declarer's partner is known as the dummy.
Example of an auction (North dealt):
North East South West pass 1 heart double 3 hearts 3 spades pass 4 spades pass pass pass
North-South will try to win at least 10 tricks with spades as trumps; North, who mentioned spades first, is the declarer. South's double of one heart was cancelled by West's bid of 3 hearts.
The player to the left of the declarer leads to the first trick and may play any card. Immediately after this opening lead, the dummy's cards are exposed. The dummy should arrange them neatly in suits, the cards of each suit arranged in rank order in an overlapping column, pointing towards the declarer, so that all the cards are clearly visible. The trump suit if any should be to dummy's right (declarer's left); in the diagram, spades are trump.
Play proceeds clockwise. Each of the other three players in turn must if possible play a card of the same suit that the leader played. A player with no card of the suit led may play any card. A trick consists of four cards, one from each player, and is won by the highest trump in it, or if no trumps were played by the highest card of the suit led. The winner of a trick leads to the next, and may lead any card.
Each trick is gathered together and turned face down when complete, but you may ask to see the cards and ask who played which card until you or your partner has played to the next trick. The tricks won are to be arranged neatly in front of one member of the winning side, so that they can easily be counted.
Dummy takes no active part in the play of the hand. Whenever it is dummy's turn to play, the declarer must say which of dummy's cards is to be played, and dummy plays the card as instructed (provided that it is legal). Dummy is not permitted to offer any advice or comment on the play. When dummy wins a trick, the declarer specifies which card dummy should lead to the next trick. If when calling for a card the declarer specifies the suit only, dummy is to play the lowest card of that suit.
It is also legal, and not unusual, for the declarer to play dummy's cards by physically taking them from dummy's hand rather than just calling for them. This allows the dummy player to leave the table during the play of the hand.
As its name suggests, Rubber Bridge is played in rubbers. A rubber is the best of three games. A game is won by the first team to score 100 or more points for successful contracts, over several deals if necessary.
A side which has already won one game towards the current rubber is said to be vulnerable. A side which has not yet won a game is not vulnerable. A side which is vulnerable is subject to higher bonuses and penalties than one that is not.
The score is kept on a piece of paper divided into two columns headed WE and THEY, for the two teams, with a horizontal line part-way down (see example). Scores for successful contracts are entered below the line, and count towards winning a game. Other scores, such as bonuses for tricks made in excess of the contract (overtricks), or penalties for tricks short of the contract (undertricks) are entered above the line, and do not count towards winning the game.
Score for making the contract
For a successful contract, the score below the line for each trick (in excess of 6) bid and made is as follows:
- If trumps are Clubs or Diamonds, 20 per trick
- If trumps are Hearts or Spades, 30 per trick
- If there are No Trumps, 40 for the first trick, and 30 for each subsequent trick.
If the contract was doubled the above scores are doubled. If it was doubled and redoubled, they are multiplied by 4.
In addition, the declarer's side scores an extra 50 points above the line if they succeed in a doubled contract. This is sometimes known as "50 for the insult". For making a redoubled contract the bonus is 100 above the line.
Because of the difference in score, clubs and diamonds are called the minor suits and hearts and spades are the major suits.
A contract to make 12 tricks is known as a small slam. A contract to make all 13 tricks is called a grand slam. For bidding and making a slam, declarer's side get an extra bonus above the line, depending on their vulnerability, as follows:
Slam bonus small slam grand slam not vulnerable 500 1000 vulnerable 750 1500
Score for overtricks
If the declarer's side wins more tricks than were bid, and were not doubled, then in addition to the score below the line for the contract, they score for the overtricks above the line at the same rate as for bid tricks - i.e. 20 per trick if a minor suit was trumps; 30 per trick in a major suit or no trumps.
If the contract was doubled or redoubled, the bonus for overtricks does not depend on the trump suit, but does depend on whether the declarer's side was vulnerable as follows:
Score per overtrick doubled redoubled not vulnerable 100 200 vulnerable 200 400
Penalty for undertricks
If the declarer's side win fewer tricks than they bid, neither side scores anything below the line, but the declarer's opponents score above the line. This score depends on the declarer's side's vulnerability, and whether the contract was doubled or redoubled, as follows:
Undertrick penalty: not vulnerable vulnerable Not doubled - each undertrick: 50 100 Doubled - first undertrick: 100 200 Doubled - 2nd and 3rd undertrick: 200 each 300 each Doubled - subsequent undertricks: 300 each 300 each
Redoubled undertricks cost twice as much as doubled undertricks.
The top five trumps (A K Q J 10) are called honours. If one player holds all five of these cards, that player's side scores a bonus of 150 above the line. Four honours in one hand score 100. If there are no trumps, and a player holds four aces, that player's side scores 150 for honours.
Scores for honours are to be claimed at the end of the play (it is assumed that the players will remember what they held).
As there is no skill in scoring for honours, players often agree to play without the honour bonuses.
Game and Rubber
A side that accumulates 100 points or more below the line has won a game. A new line is drawn under the scores. Anything the opponents had below the line does not count towards the next game - they start from zero again.
It is important to notice that, starting from zero and in the absence of doubles, to make a game in one hand you need to succeed in a contract of at least three no trumps, four spades, four hearts, five clubs or five diamonds.
The side which first wins two games wins the rubber. For this they get a bonus of 700 if they won it two games to zero, or 500 if it was two games to one. Both sides' scores are then totalled and if the game is being played for money, the side with the higher score wins an amount proportional to the difference in scores from the side with the lower score.
If play ends for any reason with a rubber unfinished, then a side with a game gets a bonus of 300 points, and a side with a part score (i.e. a score below the line towards an uncompleted game) gets a bonus of 100.
Example of Rubber Bridge scoring
The scoresheet of a completed rubber might look like the example below. (The letters in brackets indicate successive deals as described in the corresponding footnotes - they would not appear on the scoresheet.)
WE | THEY | 500 (f) | 50 (f) | 100 (f) | 200 (e) | 500 (i) 300 (b) | 30 (g) 60 (a) | 30 (c) =============|============== <-- the line 60 (a) | 100 (c) -------------|-------------- (c) 360 (f) | 90 (d) -------------|-------------- (f) 60 (h) | 40 (g) | 90 (i)
(a) we bid 2 hearts and made 10 tricks - 60 below the line for the contract and 60 above for the overtricks
(b) they bid 4 spades, we doubled them, and they took only 8 tricks - we score 100 for their first undertrick and 200 for the second
(c) they bid 3 no trumps and made 10 tricks. This gives them a game (100 below the line) plus 30 above the line for their overtrick. A new line is ruled below the scores to indicate the start of a new game.
(d) they bid and made 3 spades
(e) they bid two diamonds and made 6 tricks - they are now vulnerable so we score 100 for each undertrick.
(f) we bid 6 hearts; they doubled us, but we won all 13 tricks. We score 360 (180 x 2) below the line for our doubled contract, giving us a game; 100 above for our doubled non-vulnerable overtrick; 50 above for making a doubled contract; and 500 bonus for a small slam bid and made.
(g) they bid one no trump and took 8 tricks; note that their 90 on deal (d) was part of the previous game, so the 40 below does not give them a game.
(h) we bid 3 clubs and made exactly 9 tricks.
(i) they bid 3 hearts and took exactly 9 tricks giving their second game and the rubber, for a bonus of 500 (two games to one).
Adding up the scores, we have 1690 and they have 880. Therefore we have won by 810 points (even though they won the rubber).
In this example the "above the line" scores were entered starting immediately above the line and working upwards. This is traditional, at least in Britain, but not necessary - you can start at the top, just below the WE-THEY headings, and work downwards if you prefer.
* Note on changes in scoring
Some details of Bridge scoring were changed in 1987 for Duplicate Bridge and in 1993 for Rubber Bridge. Before the changes, the penalty for doubled undertricks when not vulnerable was 100 for the first and 200 each for all others (and twice as much for a redoubled contract). Also the bonus for making a redoubled contract was 50, not 100, and the bonus for a part score in an uncompleted rubber was 50, not 100.
Partnership agreement and conventions
As in most card games, partners are forbidden to convey information to each other by talking, gestures, facial expression, etc. However there is considerable scope for partners to exchange information within the rules of the game by their choice of bids or cards played.
The bidding mechanism is such that if a player makes a bid (or double or redouble), it is always possible for the player's partner at their next turn to override that bid with a higher bid. This makes it possible for partners to assign arbitrary meanings to bids. Bids which can be taken at face value - that is they convey a genuine wish to play a contract to take the relevant number of tricks or more with the trump suit stated - are called natural. Bids which carry an agreed meaning other than this are called artificial or conventional.
For example if we are partners, we might agree that a bid of one club by me shows a strong hand, but has nothing to do with wanting clubs as trumps. Provided that we both understand this, you will not leave me to play a contract of one club, but will make some other bid, natural or artificial. Another example: since doubling a low-level suit contract in the hope of a penalty is unlikely to be profitable, almost all players use an agreement that in certain situations a double simply shows a good hand (perhaps with additional specifications) and asks partner to bid - this is known as a takeout double.
A bidding system is a comprehensive set of partnership agreements about the meanings of bids. For natural bids, players commonly have agreements on the number of cards held in a bid suit: for example in some natural systems, opening the bidding with one of a major suit implies a holding of at least five cards, while others require only four or more cards in the suit. Agreements also often relate to high cards held in the bid suit or in the hand generally. The overall strength of a hand is often measured in "high-card points", counting each ace=4, king=3, queen=2, jack=1, and 0 for other cards - an approach originally developed by and named after Milton Work in the 1920's. Most bidding systems also include some agreements that in certain circumstances a player will not pass. A bid, natural or artificial, which by agreement requires the player's partner to respond with a bid, rather than pass and allow the contract to be played, is called a forcing bid. Sometimes the agreement is that after a certain bid both partners agree to keep the auction going until it reaches a certain level. For example a bid that by agreement requires the partnership to continue bidding until a game contract is reached is known as a game-forcing bid. Such methods help ensure that with suitable cards partners will have more than one opportunity to bid. These extra bids can be used to exchange further infomation about their hands and improve their chances of reaching the best contract.
The main restriction on agreements between partners about the meaning of bids is that all such agreements must be declared to the opponents. Players should declare their system (if any) at the start of a session. Many clubs and tournaments require that this be done by means of a convention card which sets out the meanings of bids. In addition, an player may, at their turn to bid or play, ask for and be given an explanation of the opponents' bidding agreements. The explanation should be given by the partner of the player who made the bid in question. For example, if I double a suit contract, either opponent may, at their turn, ask my partner what the double means, and my partner must answer according to any agreement we may have about the meaning of the double - for example that it is for takeout or for penalties. If we have no agreement on this, partner should say so - players are not required or permitted to speculate or to guess at the meanings of bids in answer to such a question.
It is sometimes agreed that after the auction, the declarer's left hand opponent, having asked any necessary questions about the declarer's side's bidding agreements, leads the first card face down. The other opponent may then ask questions about the declarer's side's bidding, after which dummy's cards are exposed and play continues as usual. This procedure minimises the risk that by asking a question you may give unauthorised information to your partner. Asking at other times during the bidding or play, though legal and sometimes necessary, might be taken to imply that your next bid or play will depend on the answer given.
Similar considerations apply to the play. Partners may agree on the meaning of the choice of card played in certain circumstances. For example we may agree that when leading from a sequence of adjacent high cards such as K-Q-J we always lead the highest. Again, the opponents are entitled to know about such agreements. They should be declared on the convention card, and may be asked about during the play.
In Rubber Bridge one does not often come across complicated systems and partnership agreements. One is often playing with an unfamiliar partner, or in an informal setting. Complicated agreements are more often encountered in Duplicate Bridge, where the players are often long standing partners who have devoted considerable effort to agreeing their system.
In Rubber Bridge, although the better players have a noticable edge and will undoubtedly win in the long run, the outcome of a single rubber depends heavily on which side is dealt the better cards. The idea of Duplicate Bridge is to eliminate this element of luck, by having the same hands played more than once, by different sets of players.
Suppose we are partners and play a hand of Duplicate Bridge as North-South. Instead of being rewarded for our absolute score on that hand, our score is compared with those of other players who played the same deal as North-South against other opponents. We win if we score better than other players managed with our cards, and lose if we score worse.
For this comparison to be fair, it is necessary that each group of players who play the same deal should start from the same position. Therefore it is not practicable to play rubbers, where the scores carried forward from deal to deal affect the tactical situation. Instead, each deal is scored in its own right, and does not affect the scores for subsequent ones. The concept of vulnerability is retained, but on each deal the vulnerability is preassigned.
An almost essential piece of apparatus for playing Duplicate Bridge is a set of duplicate boards, and a pack of cards for each board. Each board contains four pockets marked North, East, South and West in which the cards for the four players are stored. Each board also carries a number to identify it, and has marks showing which of the players is dealer and whether each team is vulnerable or not. The marking of the boards is as follows:
- Board 1: dealer North; neither side vulnerable
- Board 2: dealer East; North-South vulnerable
- Board 3: dealer South; East-West vulnerable
- Board 4: dealer West; both sides vulnerable
- Board 5: dealer North; North-South vulnerable
- Board 6: dealer East; East-West vulnerable
- Board 7: dealer South; both sides vulnerable
- Board 8: dealer West; neither side vulnerable
- Board 9: dealer North; East-West vulnerable
- Board 10: dealer East; both sides vulnerable
- Board 11: dealer South; neither side vulnerable
- Board 12: dealer West; North-South vulnerable
- Board 13: dealer North; both sides vulnerable
- Board 14: dealer East; neither side vulnerable
- Board 15: dealer South; North-South vulnerable
- Board 16: dealer West; East-West vulnerable
After board 16 the pattern repeats - board 17 is like board 1, board 18 like board 2 and so on.
Before the boards are played the cards are shuffled, dealt and placed in the pockets. Traditionally, this was done by a neutral person or by a player in the presence at least one opponent. Nowadays the cards are often dealt by computer, with the aim of ensuring perfect randomness while enabling a record of each deal to be kept. A simple method is for the computer to produce a printed hand record or a set of curtain cards, specifying which cards should be in each hand on each board; a neutral person then has to construct the hands and put them in the pockets. Since the early 21st century, however, computer controlled dealing machines have become widely available. These machines physically sort the playing cards and place them in the boards ready to be played. Early models did this with the aid of a bar code printed on each card; these are gradually being superseded by machines that use optical character recognition (OCR) to identify and deal standard playing-cards.
When about to play a board, the players take their cards from the appropriate pockets, check to see that they have 13 each, and then bid as usual. The mark on the board showing the 'dealer' in practice just indicates which player is to begin the bidding. The opening lead is always made face down, as explained above, to give the leader's partner an opportunity to ask questions about the bidding before the led card is shown. During the play, the cards are not played in the centre of the table but in front of the players. At the end of each trick, all four players turn their played card face down. The cards played by each player are overlapped, with the longer axis of the card pointing to the winners of the trick (i.e. the cards belonging to tricks you have won are placed upright from your point of view, and the ones belonging to lost tricks sideways). That way you can easily see how many tricks you have won. Also, if the cards are kept in order, any dispute about revokes or tricks won or lost can be settled by reconstructing the play. At the end of the play, each player's cards are gathered up and replaced in the correct pocket, ready for the next time the board is to be played.
When this method of play is used, dummy is expected to remain at the table if at all possible, and declarer then always calls dummy's cards rather than pulling them from the dummy. You may ask to look at the cards played to a trick by the other players as long as your own card is face up. Once you have turned your card face down, you no longer have the right to see any of the other cards played to that trick. (Unless you are dummy, you are still allowed to peek at your own played card, without exposing it, until the lead is made to the next trick.)
Each board is marked to show whether both sides, one side or neither side is vulnerable for that board. You still need to score at least 100 points for tricks bid and made to make a game, but on each board, both sides start with zero points towards games - there are no 'part scores' carried forward.
In place of the rubber bonus, there are game and part score bonuses:
Making a game when vulnerable: 500 points Making a game when not vulnerable: 300 points Making a part score any time: 50 points
The rest of the scores are the same as in Rubber Bridge, except that there are no bonuses for honours in Duplicate Bridge. So for example:
- if we bid 2 spades and make 4 (10 tricks) we score 170, that is 60 for two spades bid and made, 60 for two overtricks and 50 for the part score;
- if we bid 4 spades and make it when we are not vulnerable we score 420 (120 for the contract and 300 for the game);
- if we bid 4 spades and make it when we are vulnerable we score 620 (120 for the contract plus 500 for the game).
These scores are of course not yet the final scores. They have yet to be compared with the scores achieved by other people who have played the same cards as us on this board. The method of doing this comparison varies according to what kind of duplicate is being played. Perhaps the commonest types are teams of four with international matchpoint (IMP) scoring, and matchpointed pairs.
Teams of Four
A match can be played between two teams of four - eight players in all. Each team consists of two partnerships, and you need two tables - preferably in separate rooms so that players cannot overhear events at the other table. Before starting the players agree how many boards will be played - this could be 24, 32, 48 or more, depending on the seriousness of the match and the time available. A 24 board match should easily be completed within three hours. Shorter matches, sometimes of as few as 6 boards, are commonly played if the match is part of a larger tournament. Longer matches are normally split into two or more segments (or stanzas) after each of which there may be a break and an opportunity to change seats.
Call the tables 1 and 2 and the teams A and B. Then the pairs of team A sit North-South at table 1 and East-West at table 2, and the pairs of team B occupy the other seats. Take a convenient number of boards - say boards 1 to 12 - and give the first 6 to table 1 and the other 6 to table 2. As each table finishes their 6 boards they are passed to the other table to be replayed. Since none of the players should go near the other table before everyone has played all 12 boards, it is best if the boards are transferred from table to table by a neutral referee; if none is available, the boards that have been played once can be left in a place away from both tables for collection by the players from the other table. When all 12 boards have been played at both tables, it is a convenient time to compare scores and maybe enjoy some refreshments.
It may be agreed that for the next segment, the two pairs of one of the teams should swap places. This gives each pair the opportunity to play against both pairs of the opposing team. The procedure about the number of segments in a match and the choice of seats for each segment may be laid down by the organiser of the event - otherwise it needs to be agreed between the team captains.
Each player should have a score card to record the score on each board. The card has a row for each board. The beginning of North's card from table 1, when completed, might look like this:
Board Final Score IMPs Deal # Vul Contract By Tricks Plus Minus Plus Minus N 1 - 4S S 10 420 E 2 NS 5D* W 8 500 S 3 EW 3NT W 12 690 W 4 All 2H N 9 140
In the contract column 5D* means 5 diamonds doubled. The 'By' column shows who was declarer. The score is recorded from player's point of view (North's in the example) - so when West goes down in 5 diamonds it is positive. The IMPs can only be filled in when this card is compared with one of the cards from the other room. Some players prefer to enter the number of over- or undertricks in the "Tricks" column rather than the total number of tricks taken. In that case the "Tricks" column entries for the four boards in the above example would read "=, -3, +3, +1". Suppose that our team mate East on table 2 has a card like this:
Board Final Score IMPs Deal # Vul Contract By Tricks Plus Minus Plus Minus N 1 - 4S S 11 450 E 2 NS 4H N 10 620 S 3 EW 6NT W 12 1440 W 4 All 4H N 9 100
Now the differences can be converted to IMPs for the team. The following standard table is used:
Point difference IMPs 0 - 10 0 20 - 40 1 50 - 80 2 90 - 120 3 130 - 160 4 170 - 210 5 220 - 260 6 270 - 310 7 320 - 360 8 370 - 420 9 430 - 490 10 500 - 590 11 600 - 740 12 750 - 890 13 900 - 1090 14 1100 - 1290 15 1300 - 1490 16 1500 - 1740 17 1750 - 1990 18 2000 - 2240 19 2250 - 2490 20 2500 - 2990 21 3000 - 3490 22 3500 - 3990 23 4000 or more 24
So in the example, on the first board the difference between the two tables was 30 against us, and we lose 1 IMP. On the second board we lose 3 IMPs. Although on table 1 our North-South pair defeated West's 5 diamonds, on table 2 with the same cards our East-West pair allowed North to play and make 4 hearts. On board 3, where we bid the small slam on table 2, while they stopped in game on table 1, we gain 13 IMPs for a 750 point difference. On board 4 both Norths made 9 tricks in hearts, but we gain 6 IMPs because our North-South pair just bid 2 hearts rather than 4. Overall we are 15 IMPs ahead on those four boards.
After each scoring interval, the captains of the teams should check that the scores agree. The purpose of every player keeping score is to make it easier for errors to be traced and corrected.
At the end of the match, the result is the difference in IMPs between the teams. Sometimes there is then a further conversion of this margin into a match result, in which some fixed number of victory points is apportioned between the teams. There is no single standard conversion table, but here is an example table for a 24 board match:
IMP difference Victory Points 0 - 2 10 - 10 3 - 6 11 - 9 7 - 11 12 - 8 12 - 16 13 - 7 17 - 21 14 - 6 22 - 27 15 - 5 28 - 33 16 - 4 34 - 39 17 - 3 40 - 46 18 - 2 47 - 54 19 - 1 55 or more 20 - 0
In the example, if we were still 15 IMPs ahead having played 24 boards, using this table we would win the match 13-7. If the match was part of some larger competition, such as a league, then we would score 13 victory points and our opponents would score 7.
There are also events in which many teams of four compete. There are various ways of organising these. At any particular time in such an event you will be playing a part of a match against some other team, and at some time your team-mates will play the other cards of the same boards against the other half of that same team. The scores are eventually compared to find how many IMPs you won or lost against that team.
Another way of scoring teams of four is akin to the matchpoint scoring used in pairs (see below). On each board you simply win, tie, or lose depending on whether you score better, worse or the same as the other team. This method is known as board-a-match or BAM in America; in Britain it is usually called point-a-board.
This is the game most usually played in Bridge clubs, and there are also many tournaments organised this way. As implied by the name, it is played between a number of fixed partnerships or pairs. For a pairs event you need a minimum of three tables (6 pairs, 12 players), and it works better with more players - say 10 tables (40 players) or more. With a very large number of players (say more than 70) it is usual to split the tournament into two or more separate sections.
Generally you play two or three boards at a table - this is called a round - and then one or both pairs move to another table and play other boards against other opponents. The movement will be organised by the director in such a way that no one ever plays boards they have played before, or against opponents they have played before.
Traditionally, the score for each hand was recorded to a travelling scoresheet or traveller, which was kept in the board, folded so that previous scores could not be read, either in a special pocket provided for this purpose, or in the North pocket on top of North's cards. None of the players may look at this sheet before the board has been played. North is then responsible for entering the result and showing the completed sheet to East-West to check that it has been done correctly. Each pair has a number to identify them, and this must also be entered on the scoresheet, to show whose result it is. North is also responsible for the movement of the boards - checking at the start of the round that the correct boards are being played and passing them on at the end of the round.
At the end of the whole session, each scoresheet will contain the results of all the pairs who have played that board. The scoresheets are then collected by the organisers and the scores compared. The usual method of scoring is in matchpoints. Each pair is awarded 2 matchpoints for each pair who scored worse than them on that board, and 1 matchpoint for each pair who scored equally. (In North America it is customary to count just one matchpoint for each pair scoring worse than you on a board, and half a matchpoint for those that are equal. This obviously makes no difference to the final ranking order or percentages scored by the pairs. Half points are traditionally written as a horizontal or diagonal stroke: — or /. )
A completed score sheet might look like this:
Board No. 1 Pair No. North-South Matchpoints NS EW Contract By Tricks Plus Minus NS EW 1 8 4S N 10 420 5 7 2 13 3NT S 10 430 8 4 3 11 5C* E 8 500 12 0 4 9 4S N 10 420 5 7 5 14 4S N 11 450 10 2 6 12 5S N 10 50 0 12 7 10 3S N 10 170 2 10
Then the total matchpoints scored by each pair over all the boards are calculated. This is generally converted to a percentage for each pair of the points they scored compared to the theoretical maximum. This gives a fair comparison between pairs who have played different numbers of boards. The winners are the pair with the highest percentage. There may be prizes for 1st, 2nd, 3rd place, etc.
Another, less usual way of scoring pairs is with a version of the IMP scoring used for team matches (see above). There are two kinds of IMP pair games: your score may be IMPed against every other pair that played the same hands, or against a form of average of the scores of all the pairs who played the hand.
Sometimes the movement is such that the North-South pairs stay put and the East-West pairs remain East-West throughout. In this case the results for the East-West pairs and the North-South pairs are separate, and there are two winning pairs. To enable all the pairs to be placed in a single ranking order, the last round is sometimes played with an arrow switch. This means that the players who were previously North-South play the East-West cards for that round and vice versa.
I am told that in many North American tournaments "pickup slips" were used instead of travellers - that is, there was an individual score slip for each table in each round. After North had filled out the details and East or West had checked it, it was left face down on the table and picked up by the organisers during the next round. This facilitated the calculation of final scores by computer, as the results of earlier rounds could be entered while the later rounds are being played. It also prevented players from seeing the results obtained by other players who previously played the same cards, which might be considered an advantage or a disadvantage.
Nowadays it is increasingly common for scores to be entered directly into a computer terminal at the table. These can be configured to provide or not to provide information about previous results for that board, and can be set to ask the person entering the details to also record the opening lead.
Duplicate Bridge procedure and ethics
During a duplicate event, where play will be in progress at several tables at the same time, it is important that players do not see, overhear or otherwise take an interest in the play at the other tables. Any attempt to do so would be cheating, as it might give unauthorised information about the distribution of cards or the result of a board which the player would later be playing. For similar reasons, partners should not discuss the boards they have played in the hearing of other players until the end of the event (or a suitable break at a time when everyone has played the same boards).
In many places devices are used to enable the bidding to proceed silently, reducing the chance of hearing bids from another table. The best arrangement is for each player to have a bidding box, which is a box containing cards displaying all the possible bids, pass, double and redouble. At your turn you display the relevant card. All the cards used for bids remain on view until the end of the auction, thus also avoiding the problem of players forgetting or mishearing part of the bidding. A cheaper but less satisfactory method is to use a large card with a compartment for each possible bid; at your turn you point to the bid you wish to make. I am told that in Australia, overhearing of bids is commonly avoided by requiring bids to be written down rather than spoken.
In an event of any size, there will be a tournament director whose job is to ensure that the play flows smoothly. This person will deal with any infringements of the rules that occur, referring when necessary to the laws. If some irregularity occurs, such as a bid out or play out of turn, an illegal bid or play, or discovering that the cards have been wrongly boarded (the hands contain more or fewer than 13 cards), the director should be called to the table. This should not be construed as an accusation of cheating - the purpose of calling the director is simply to ensure that the irregularity is sorted out fairly and in accordance with the rules. The instructions and decisions of the director should be followed and respected at all times. In a serious tournament, if you strongly disagree with the director's ruling, it should be possible to appeal against the director's decision. The procedure for this varies according to the nature of the event - the director should be able to advise you on the options.
Stop and alert
In tournament Bridge, if you make a bid at a level higher than necessary in that denomination (a "jump" bid), you are supposed to precede your bid by saying "stop" (or displaying your "stop" card if you are using bidding boxes). The next player must then pause before bidding or passing. The reason behind this is that after a jump bid the next player may have reason to hesitate, as your unexpectedly high bid might have disrupted the course of action which that player was planning. The player is forced by the stop rule to hesitate anyway, so avoiding giving unauthorised information. Example:
- North bids "one spade"
- East bids "stop; three hearts"
- South pauses and then passes
If South had been planning to bid two spades, say, then he might need time after East's unexpected jump to decide whether a three spade bid would now be appropriate. As South is forced to pause, North gets no clue as to whether the jump gave South a problem. Similarly, if South instead bids three spades after the mandatory pause, he gives North no clue as to whether he was considering a pass instead.
I am told that in North America the correct use of "stop" cards is poorly understood by players and that for this reason from the start of 2018 "stop" cards will no longer be used in tournaments, and will probably also be abandoned by bridge clubs. In other parts of the world, for example in Britain, "stop" cards continue to be used.
The idea of alerts is to warn the opponents of a bid (or double or pass) which has an unexpected agreed meaning. It is always the duty of the partner of the bidder to alert the bid when required. If using bidding boxes, this is done by displaying the "alert" card. Otherwise the alert is given by saying "alert" or (in Britain but not in North America) by knocking the table. The definition of what bids require alerts varies from place to place - it is determined by the Bridge organisation under whose aegis the tournament is being held. In Britain, most artificial bids must be alerted; in North America, alerts are required for bids which diverge from a defined standard set of meanings.
Since the late 1990's, "announcements" have been introduced in some places. When bids with certain specific meanings are made, the bidder's partner must say a specific phrase that explains the meaning of the bid. For example the partner of a player who makes an opening bid of "one no trump" might be required to disclose the partnership's agreed range of strength for that bid in "high-card points", by saying for example "12 to 14" or "16 to 18".
This is information which you obtain in some other way than as a legitimate deduction from the bidding and play. Unauthorised information might arise from:
- hesitation or undue haste in bidding or playing a card
- seeing your partner's cards
- extraneous remarks made during the game; also gestures, tone of voice, etc.
- seeing or overhearing events at another table
- questions your partner asks about the bidding or play
- alerts/announcements made or not made by your partner, or answers to opponents' questions, if they reveal that you and your partner differ as to the agreed meaning of your bids or plays
The principle is that you are allowed to take advantage of anything done by your opponents at your table, but you are obliged to ignore any unauthorised information gained from your partner's actions or from other tables.
In fact if you do obtain unauthorised information from your partner, you should not only ignore it but be prepared to prove that you have done so. This means that if you are involved in any kind of close decision you ought to take the action opposite to the one indicated by the information from your partner. For example if during the bidding your partner passes after a hesitation, you must pass too unless you have a cast iron case for bidding, otherwise you might be accused of making use of the unauthorised information that your partner had nearly enough strength to bid.
In Bridge it is illegal to behave deliberately in such a way as to try to give spurious information to the opponents. For example if you have only one card of a suit that is led, it is illegal to hesitate before playing it, creating the impression that you had more than one card to choose from. (Even an inadvertent hesitation would be an offence, though a less serious one, if it misled the opponents to your benefit, and the director would adjust the score to give a fair result.) On the other hand there is no ban on making deceptive bids and plays to confuse the opponents - as long as these are not part of an undisclosed partnership agreement. You are free for example to play a card different from what might be expected from your holding, provided that you play the card smoothly and without comment. Similarly you are free to make a bid which is inconsistent with your system to upset the opposition, provided that this is as much of a surprise to your partner as it is to the opponents. Of course you must always bid and play legally, in turn and in accordance with the ranking of bids, the rules of following suit, and so on, even if your choice of bid or play is unorthodox and unexpected.
There are several versions of this game, also known in the official rules as Four-Deal Bridge. As this name suggests it is a game for four players which is complete in four deals, unlike Rubber Bridge, where the length of a rubber is indefinite. This greater predictability has made it popular in some American clubs where Rubber was formerly played.
The vulnerability varies from hand to hand in a fixed pattern as follows:
- Hand 1: Dealer North; neither side vulnerable
- Hand 2: Dealer East; North-South vulnerable
- Hand 3: Dealer South; East-West vulnerable
- Hand 4: Dealer West; both sides vulnerable
If all four players pass, the cards are shuffled again and the hand redealt by the same dealer. The game bonus is 500 when vulnerable, 300 when not vulnerable. If a team makes a part score this is carried forward to subsequent deals until one side makes a game. If a team makes a part score in hand 4 that is not sufficient to complete a game, they score a bonus of 100, but there is no bonus for any part scores made in earlier hands.
Example of scoring
N-S | E-W | 100 (d) | 50 (d) | 200 (d) | 500 (c) 60 (a) | 30 (c) ===========|=========== 1. N deals, love all 60 (a) | | 2. E deals, N-S vul | 40 (b) | 3. S deals, E-W vul | 60 (c) -----------|----------- 4. W deals, game all 80 (d) | ==================== Totals 550 630 so E-W win by 80.
(a) On hand 1, N-S bid two spades and made 10 tricks, scoring 60 points below the line plus 60 above for the overtricks. No game here because N-S's score below the line is less than 100.
(b) On hand 2, E-W bid one no trump and made exactly seven tricks, for 40 points.
(c) On hand 3, E-W bid two hearts and won 9 tricks for 60 points below and 30 above. This brings their score below the line to 100, which counts as a vulnerable game because E-W are vulnerable for this hand, even though this is the first game they have made.
(d) On hand 4, the final hand, N-S bid one no trump; although they are doubled, they succeed in making eight tricks. This is worth 80 (2*40) below the line, and above the line they score 200 for one vulnerable overtrick plus 50 for making a doubled contract. However, this is not a game: their 60 below the line from hand 1 no longer counts towards game because of the game E-W made in hand 3. Therefore N-S score just a further 100 for finishing with a part score.
Note: the original version of Chicago had the vulnerability reversed in hands 2 and 3, so that the dealing side was vulnerable. The more modern scheme, which has the non-dealing side vulnerable as shown above, tends to lead to more competitive bidding.
Chicago with Duplicate Scoring
Chicago is sometimes played using duplicate scoring. There is no accumulation of part scores or games from deal to deal - each deal is scored separately, and a team making a part score gets an immediate bonus of 50 as in duplicate. The sequence of vulnerability is fixed as in the standard version.
A multiple of four hands can be played, repeating the sequence of vulnerabilities as often as necessary. The result is simply the total score over the deals played.
Chicago with Russian Scoring
The following method of scoring Chicago originated in Russia. It eliminates some of the luck of the deal by introducing an element of IMPs scoring.
On each deal, there is a target score which depends on the number of high card points held. The cards are played in front of the players, as in duplicate. At the end of the play, the high card points held by each side are counted, according to the following scale:
- each ace: 4 points
- each king: 3 points
- each queen: 2 points
- each jack: 1 point
There are 40 points in all. The team which held more high card points finds its target score, which depends on whether they were vulnerable or not, from the following table:
High Card Target Points Not Vul. Vulnerable 20 0 0 21 50 50 22 70 70 23 110 110 24 200 290 25 300 440 26 350 520 27 400 600 28 430 630 29 460 660 30 490 690 31 600 900 32 700 1050 33 900 1350 34 1000 1500 35 1100 1650 36 1200 1800 37 1300 1950 38 1300 1950 39 1300 1950 40 1300 1950
The difference between the target score from the above table and the actual score is then converted to IMPs, using the standard IMP table. The total IMP scores over a series of hands are totaled to give an overall result.
For example, suppose we are East-West, and on the second deal of a Chicago we bid three hearts and make 10 tricks. We then count our high card points and discover that between us we had 24. We were vulnerable, so our target score from the table was 290. We actually scored 170 (90 for the contract plus 30 for the overtrick plus 50 for the part score). So we are 120 points short of our target. Therefore using the IMP table, our score for this hand is minus 3 IMPs.
Second example. In the first deal of a Chicago we bid and make 4 Spades holding only 18 points between us. Our opponents had a target of 70 but instead we made 420. The difference is 490 so we score plus 10 IMPs.
There are several alternative versions of this scoring table. In Estonia, a compensation table is used which also takes into account the fit between the hands of the partners with the majority of high card points. The details are available on Tanel Teinemaa's Compensation Table web site.
The Beer Card
The Beer Card is the Seven of Diamonds. It is not part of the official rules of Bridge, but there is a tradition among some players that if the declarer succeeds in making the contract and wins the last trick with the Seven of Diamonds, dummy must buy the declarer a beer of the declarer's choice. In the same way, if the opponents defeat the contract and one of them wins the last trick with the Seven of Diamonds, the opponent who wins the last trick is bought a beer by the other opponent.
The Beer Card tradition originated in Copenhagen in the 1950's or 1960's. It was probably inspired by:
- the large reward for winning the last trick with a King or the Pagat (lowest trump) in the game of Danish Tarok, or the bonus for winning the last trick with the trump 7 (the lowest trump) in the Danish form of Skat;
- the fact that the diamond seven is a valuable card in the system of bommelommer points - a way of evaluating a Bridge hand which has little or no connection with its usefulness in the game of Bridge, but was used in some Danish clubs as the basis of a side-bet between partners. Bommerlommer is a slightly old-fashioned Danish slang word for money.
Bridge has become so popular and fashionable that some players can hardly believe that any other card game is worth learning, but Bridge is a four-player game. When two such people want to play cards and no other players are available, instead of playing a card game designed for two players, they sometimes prefer to resort to two-player adaptations of Bridge, known as Honeymoon Bridge. There are several different versions, all somewhat unsatisfactory. Rules can be found on the Honeymoon Bridge page of this site.
Minibridge is a simplified version of Bridge that was introduced in Europe in the 1990's as a teaching aid for new players. In its most straightforward form it works as follows.
After the cards are dealt as usual, the players look at their cards and count the number of high card points they hold according to the following scale: ace=4, king=3, queen=2, jack=1, other cards zero. This is the popular Milton Work point count used by many Bridge players to evaluate the approximate strength of a hand for bidding purposes. But instead of bidding in the usual way, in Minibridge each player in turn, beginning with the dealer, simply announces his or her point count. The point counts of the four players should add up to 40. The partnership with the higher total point count plays the contract, the declarer being the whichever player of that partnership holds more points. If each partnership has 20 points there is a redeal by the same dealer. If both members of the declaring partnership have the same number of points (for example 12 each), the declarer is the member of the partnership who spoke first - that is the dealer or the player to dealer's left.
The declarer's partner puts down the dummy, and the declarer, having seen partner's hand announces whether the contract will be a "game" or a "part score" and also the trump suit or "no trumps". Then the player to declarer's left leads to the first trick and play proceeds as in normal Bridge.
The scores for each trick above six are as usual: 20 if trumps were clubs or diamonds, 30 if trumps were hearts or spades, and 40 for the first trick and 30 for each subsequent trick if there are no trumps.
- If the contract was "part score", it succeeds if the declarer's side wins at least seven tricks. The winning side scores the trick score plus a bonus of 50 points.
- If the contract was "game", it succeeds if the declarer's side wins enough tricks to score at least 100 trick points - in other words at least 11 tricks if clubs or diamonds are trumps, at least 10 tricks if hearts or spades are trumps, or at least 9 tricks if there are no trumps. In this case the declarer's side scores the trick score plus a bonus of 300 for the "game".
- If the contract fails, the declarer's opponents score 50 points for each trick by which the declarer was short of the number needed to make the contract.
- Part score in no trumps. For 7 tricks declarer's side would score 90. For 10 tricks they would score 180. if they took 5 tricks the other side would score 100.
- Game in hearts. For 10 tricks declarer's side would score 420. For 11 tricks they would score 450. If they took 7 tricks the other side would score 150.
There is no set overall target score. Players play an agreed number of deals after which scores are compared to give the result.
There are several more elaborate versions of Minibridge played in various parts of Europe. For example instead of players announcing their points, each player writes on a slip of paper the number of points and the number of cards held in each suit. Then players speak in turn, starting with the dealer, the options being "pass" or "I open". If all pass the cards are redealt. If a player opens, the opener's partner's slip is passed to the opener, and the opener uses this to choose a contract, which is like a bid in Contract Bridge - a number of tricks above six and a trump suit or no trumps. After this, the opener's left-hand opponent receives his or her partner's slip and can either pass or "overcall". If the opponent passes the opener's contract is played. If the opponent overcalls, he or she must name a contract higher than the opener's bid: either more tricks or the same number of tricks in a higher denomination. The opener can then bid again, and the overcaller and opener continue to bid alternately, each bid being higher than the last, until one of them passes. The final bidder becomes the declarer and plays the final contract.
Further information about Minibridge can be found on the English Bridge Union's Minibridge pages.
Abridged is a proprietary game, based on Minibridge.
Bridge resources on the Internet
Rules, advice and Bridge education
Bridge is one of the few card games with official rules. Here links to
- the World Bridge Federation's Laws of Duplicate Bridge (revised in 2017)
- the Rubber Bridge Laws (2014) as promulgated by the American Contract Bridge League, and
- the Laws of Online Bridge (2001).
The Bridge World home page has some good introductory material for beginners, as well as problems, a book list, samples from their magazine, and links to other sites.
The BridgeHands site provides an indexed encyclopedia of Bridge terms, summaries of popular bidding systems, a copy of the laws, some book reviewed and other resources.
Bobby Wolff's daily Bridge column The Aces on Bridge is available to read online.
David Stevenson's Bridge Page has a collection of articles, stories, information and useful links.
The Bridge Base site provides articles and educational software and well as an on line Bridge server.
The web site of the Pattaya Bridge Club in Thailand has a fair amount of Bridge information, including a useful reference section on conventions.
Ron Klinger Bridge, the web site of the well-known Australian player, provides news, problems, advice and other resources.
Jason Feldman and Gavin Wolpert's Bridge Winners site features Bridge articles, problems and polls, tournament news, an ACBL convention card editor and a social networking framework.
The 35 Steps is a graphic visual guide to bidding on a single sheet. There are different versions for two popular bidding systems: Standard American (15-17 No Trump, 5-card majors) and Acol (12-14 No Trump, 4-card majors). It is available in various formats: A4 laminated sheet, large wall chart, mouse pad, fridge magnet, etc.
The Bridge Today University provides Bridge lessons by e-mail.
Warren's Free Bridge Workshop offers instruction based on "Standard-American Goren Party Bridge" techniques.
This archive copy of Zar Petkov's former Zar Points web site explains his hand evaluation method and the theory behind it. It included a calculator which compared valuations of pairs of hands using the Zar, Bergen and Goren point counts.
Andrei Sharko's book The Symmetric Relay Contract Bridge Bidding System Made Easy is available here. Symmetric Relay is a strong club bidding system first published in 1980 by Walter Jones and Roy Kerr. Andrei Sharko's book presents a refined version of this system in easily accessible form.
Maggy Simony's blog Bridge Table Chronicles aims to revive the popularity of Bridge by promoting it as an informal, sociable game.
The Bridge 7 site offers Bridge tutorials and software in English and in Turkish.
The Bridge Doctor site, run by New Zealand players Graeme Tuffnell and Tina McVeigh, provides on line Bridge tutorials and games.
An introduction to Rubber Bridge is available at the Card Game Heaven web site.
LeBridge.info is a French language website offering an introduction to Bridge for beginners.
Sites for playing Bridge online
- Bridge Base
- Bridge Club Live offers a 30 day free trial.
- Game Zone
- PlayOK (formerly known as Kurnik)
- Rubber and Duplicate Bridge games are included in the Favorite Games Ltd. package. A Bridge game for Android OS is also available.
- At Pierre-Marie Petit's Jeubridge site (French language) you can play Bridge tournament deals on line.
Bridge Baron - three-time winner of the world computer Bridge championship; choice of bidding systems and conventions; choice of random (recreatable) deals, deals to specified profile or preset deals; online option in Windows version.
The collection HOYLE Card Games for Windows or Mac OS X includes a Bridge program, along with many other popular card games.
BiddingQuest is a free online system through which you can practise bidding with your partner by bidding predealt hands and scoring the results, and also take part in a bidding contest.
Neural Play has produced a Bridge app for Android.
Funbridge is a bridge app for iPhone, iPad, Mac, Windows PC, Android and Amazon.You are South, and the other three hands are bid and played by the AI. You can compare your results with those of other users, and you can challenge a friend to a match in which you each play the same deals with the same cards. There is a choice of bidding systems and tournament formats.
Norm Radder's improved version of Stephen Han's EasyBridge program is available free from BoardGameGeek.
Special K Software has software to play the card games of Bridge. This software is available at www.specialksoftware.com.
From Dave Cullen's Acol Bridge site you can download WolfBridge, a freeware Bridge program for Windows.
The GOTO Bridge program includes a large number of predealt and preplayed hands, so that you can compare your performance with others.
Other WWW pages with useful Bridge links
The American Contract Bridge League has many useful resources and links.
Jude Goodwin-Hanson maintains a page of Great Bridge Links
Bridge Equipment Suppliers, Clubs and other Bridge related pages
The Baron Barclay Bridge Supplies site has an on-line catalog of their equipment and links to other Bridge pages.
Master Point Press is a publisher specialising in Bridge books.
The Mindracer site has a range of Bridge products and a bidding quiz.
The Bridge page at startkabel.nl has a collection of links to Dutch Bridge clubs, tournaments and other information.
First for Bridge offers a range of Bridge Holidays.
Hamilton Bridge Center - ACBL affiliated Bridge club in Hamilton, Ontario.
Thanks to the many people whose feedback has helped me to improve this page - especially to Mark Brader, for his many corrections and suggestions.