C.04.4.3 Lim System

Approved by the General Assembly of 1987.
Amended by the 1988, 1989, 1997, 1998 General Assemblies and 1999 Executive Board.

General Pairing Rules

Awarding the pairing-allocated bye

1.1

In addition of what is stated in the basic rules (C.04.1.c), the pairing-allocated bye is awarded to the player with the lowest rank in the lowest score-group.

Pairing a Score-group

2.1

Two players who have not yet played each other are said to be compatible provided that the pairing will not require either player to have the same colour in three successive rounds, or to have three more of one colour than the other.

2.2

The players with the same score form a score-group. The Median Score-group is the score-group with players having the score equal to half the number of rounds that have been played. Pairing begins with the highest score-group and proceeds downward until just before the Median Score-group, then continues with the lowest score-group and proceeds upwards to the Median Score-Group which is paired last. The Median-Score-group is paired downward.

2.3

Before the players in a score-group are paired, the players in the score-group who have no suitable opponents for the following reasons are identified and transferred to a neighbouring score-group:

the player has already played all the players of his score-group; or

the player has already received two more of one colour over an equal allocation and there is no compatible opponent available in the score-group to enable him to have a permissible colour; or

the player has already received the same colour in the previous two rounds and there is no compatible player in the score-group to enable the player to have the alternate colour; or

it is necessary to make even the number of players in the score-group

Such a transferred player is described as a floater. Rules on how to select the floater, if a choice is available, are given in the section on “Floater Selection Rules” (section 3).

2.4

The players in a score-group, after transfer of players where necessary, are arranged in the order of their pairing numbers and the players in the top half are tentatively paired with the players in the bottom half. These pairings are said to be proposed pairings, to be confirmed after scrutiny for compatibility and proper colour. If the players in a score-group are numbered : 1, 2, 3 … n, then the proposed pairings are (ignoring colours): 1 v (n/2 + 1), 2 v (n/2 + 2), 3 v (n/2 + 3) … n/2 v n.

2.5

Where a proposed pairing would result in the pairing of players who have already played each other, the lower numbered player of the two is exchanged for another within the same score-group. Further exchanges of opponents may be made to allow alternation or equalisation of colours where possible. How players are exchanged is described in the “Exchange Rules” (section 4).

2.6

Pairing a blocked median score-group
If the median score-group cannot be paired it should be extended step by step under the following rules:

if the number of floaters from higher score-groups is larger than the number of floaters from lower score-groups the next pairing of the lower score-group shall be cracked and the players of this pairing shall be treated as additional floaters from the lower score-group. Then the pairing of the median score-group is started again.
if the above condition is not fulfilled, then the next pairing of the higher score-group shall be cracked and the players of this pairing shall be treated as additional floaters from the higher score-group. Then the pairing of the median score-group is started again.

Floater Selection Rules

3.1

The “floater” is a player who is transferred to another score-group in accordance with Rule 2.3, or because a compatible opponent cannot be found for the player in spite of exchanges in the score-group.

3.2

When pairing proceeds downward, the floater is transferred to the next lower score-group. When pairing proceeds upwards, the floater is transferred to the next higher score-group.
When making even a score-group, determine the due colours of the players and select as the floater a player who would tend to equalise the number of players due different colours.
(In Maxi-tournaments, when pairing downward, the difference in rating between the chosen player and the lowest numbered player in the score-group must differ by 100 points or less, otherwise the lowest numbered player in the score-group is chosen as the floater. When pairing upwards, the difference in rating between the player chosen and the highest numbered player in the score-group must differ by 100 points or less, otherwise the highest numbered player is chosen as the floater.)
If the number of players due white equals the number of players due black, the lowest numbered player is chosen as the floater when pairing downward, and the highest numbered player is chosen as the floater when pairing upwards.

3.3

If there is a choice as to which player floats to a lower score-group, the player chosen is the lowest numbered player in the score-group who has a compatible opponent in the lower score-group, after excluding the opponents of other floaters who have higher scores or higher pairing numbers than the proposed floater.

3.4

If there is a choice as to which player floats to a higher score-group, the player chosen is the highest numbered player in the score-group who has a compatible opponent in the higher score-group, after excluding the opponents of other floaters who have lower scores or lower pairing numbers than the proposed floater.

3.5

If a proposed floater has no compatible opponent in the adjacent score-group, he shall, if possible, be exchanged for another player in his score-group; otherwise he shall be floated to a further score-group.

3.6

When pairing a group that includes floaters from a higher score-group, the floater with the highest score is paired first, or the floater with the highest pairing number, if scores are equal.

3.6.1

When pairing a group that includes down-floaters (DF) from a higher score-group, the floater with the higher pairing number is paired first.

3.6.2

When pairing a group with DF coming from different higher score-groups, the floater coming from the highest score group is paired first (not always the one with the highest pairing number).

3.6.3

When there are DF and UF (up-floaters) in the same score-groups (this should normally happen in the median score-group) in the upper half of score-groups or in the median group, first pair the DF, then the UF and finally the remaining players.

3.7

When pairing a group that includes floaters from a lower score-group, the floater with the lowest score is paired first, or the floater with the lowest pairing number, if scores are equal.

3.7.1

When pairing a group that includes UF from a lower score-group (in the 2nd half) the floater with the lowest pairing number is paired first.

3.7.2

When pairing a group that includes UF coming from different lower groups, the UF coming from the lowest score-group is paired first (not always the player with the highest pairing number).

3.7.3

When there are UF and DF in the same score group in the second half of score-groups, first pair the UF, then the DF, and finally the other remaining players.

3.8

When pairing downward, the floater is paired with the highest numbered player available who is due the alternate colour (provided, in Maxi-tournaments, that the ratings of proposed opponents who are exchanged for this purpose differ by 100 points or less). When pairing upwards, the floater is paired with the lowest numbered player available who is due the alternate colour (provided, in Maxi-tournaments, that the ratings of proposed opponents who are exchanged for this purpose differ by 100 points or less).

3.9

Due to their origin and their compatibility in the adjacent score-group there are 4 types of floaters listed in descending order of disadvantages.

a floater who has already floated to the score-group just being handled and has no compatible opponent in the adjacent score-group.

a floater who has already floated to the score-group just being handled and has a compatible opponent in the adjacent score-group.

a floater who has no compatible opponent in the adjacent score-group.

a floater who has a compatible opponent in the adjacent score-group.

If there is a choice the floaters should be chosen to minimise the disadvantages using the following priorities:

avoid floater(s) of type a

avoid floater(s) of type b

avoid floater(s) of type c

3.10

A floater who has floated the round just before shall not be floated due to section 2.3.d provided:

this will not produce other floaters of the types a, b, c of section 3.9
this will not decrease the number of pairings of that score-group

Exchange Rules

4.1

The proposed pairings of players obtained according to Rule 2.4 are scrutinised in turn for compliance with the compatibility statement (see 2.1). And,

when pairing downward, scrutiny of proposed pairings begins with the highest numbered player; if the pairing is found not to comply with 2.1, the lower numbered player is exchanged until a compatible pairing is found; or,

when pairing upwards, scrutiny of proposed pairings begins with the lowest numbered player; if the pairing is found not to comply with 2.1, the higher numbered player is exchanged until a compatible pairing is found.

4.2

In the following example of a score-group with six players, and pairing downward, the attempt is first made to find a compatible opponent for Player #1, the highest numbered player in the score-group.
Six players in a score-group with proposed pairings as follows:
1 v 4
2 v 5
3 v 6
If the pairing 1 v 4 is not compatible, for example, because the players had met in an earlier round, the positions of Player #4 and Player #5 are exchanged so that we have:
1 v 5
2 v 4
3 v 6
If the pairing 1 v 5 is also not compatible, a further exchange is made. The original proposed pairing and possible exchanges made to find a compatible opponent for Player #1 are as follows:
Proposed Pairing (col. 1) and Possible exchanges to find compatible opponent for #1

1 v 4
2 v 5
3 v 6

1 v 5
2 v 4
3 v 6

1 v 6
2 v 4
3 v 5

1 v 3
2 v 5
4 v 6

1 v 2
3 v 5
4 v 6

4.3

After a compatible opponent, for example, #6, has been found for Player #1, the proposed pairing for Player #2 is scrutinised. Exchanges to find a compatible opponent for Player #2 are as follows:
Proposed Pairing (col. 1) and Possible exchanges to find compatible opponent for #2

1 v 6
2 v 4
3 v 5

1 v 6
2 v 5
3 v 4

1 v 6
2 v 3
4 v 5

1 v 3
2 v 6
4 v 5

1 v 2
3 v 5
4 v 6

4.4

The exchanges to find a compatible opponent for Player #2 must at the same time leave Player #1 with a compatible opponent. If this cannot be done, for example, if Player #1 and Player #2 have previously played each other and all the other players except Player #6, then the original pairing of Player #1 with Player #6 is retained and Player #2 is floated. And,

if the score-group originally had uneven members and the lowest numbered player was floated to make even the number of players in the score-group, #2 is exchanged with the floater, originally #7 in the score-group, or,

if the score-group was originally even, then the lowest numbered player remaining must be floated in company with #2 to maintain an even number of members in the score-group.

Colour allocation rules

5.1

Where possible, and by means of exchanges, each player shall be given the alternate colour; at the end of each even-numbered round each player shall have had an equal number of whites and blacks. Moreover,

no player shall be given the same colour in three successive rounds, and

no player shall be given three more of one colour than the other.

5.2

After the first scrutiny and exchanges necessary to establish that all pairings in a score-group are new pairings, a second scrutiny with exchanges where necessary is undertaken to give each player, if possible, the alternating colour and at the same time, the equalising colour.

5.3

If one of the players in a pairing had the same colour in the previous two rounds, he must be given the alternating colour. If both players had the same colour in the previous two rounds and compatible opponents in the score-group are not available, then one or both players must be floated.

5.4

If both players in a pairing had the same colour in the previous round, then the colours they had in earlier rounds, going back in sequence, shall decide who is given the alternate colour. If players in the median score-group or above had identical histories, then the higher ranked is given the alternate colour, or, in even-numbered rounds, the equalising colour. If the players below the median score-group had identical histories, then the lower ranked player is given the alternate colour, or, in even numbered rounds, the equalising colour.

5.5

In the odd-numbered rounds, whenever possible, each player shall be given the colour which gives him one more only of one colour than the other.

5.6

In the even-numbered rounds, whenever possible, each player shall be given the colour that gives him an equal number of whites and blacks.
When both players of a pairing are due the same equalising colour, and further exchanges are not possible, the colour history will decide who is given the equalising colour, as in Rule 5.4. One player will then have two more of one colour than the other colour.
This is allowed but care must be taken not to violate Rules 5.1(a) and 5.1(b), and to equalise the player’s colours at the earliest opportunity.

5.7

(In Maxi-tournaments, an exchange of opponents to find, for example, one who is due the alternate colour is allowed only if the ratings of the opponents to be exchanged differ by 100 points or less.)

Exceptions applicable to the last round.
In the last round, the general principle C.04.1.e, requiring players with the same score to be paired if they had not met in an earlier round, shall have priority over alternation and equalisation of colours, even if it is necessary for one of the players to be given the same colour for the third round in succession, or to be given three more of one colour than the other.

Brief examples of pairing

Pairing Round One

7.1

If the number of players is uneven the lowest rated player in the Pairing List is given the pairing-allocated bye.

7.2

The colour to be given to Player #1 is decided by drawing lots; the other odd-numbered players in the upper half of the Pairing List are then given the same colour as Player #1. Player #2 together with the other even-numbered players in the upper half of the Pairing List are given the other colour.
Depending on the draw, the pairings for the first round in a tournament of forty players would be either 1 v 21, 22 v 2, 3 v 23, 24 v 4, … 40 v 20; or 21 v 1, 2 v 22, 23 v 3, 4 v 24 … 20 v 40, where the player having white is mentioned first. This is the only occasion when colours need be decided by lot.

7.3

Players who have won their games are each awarded one point; each of those who have drawn receives 0.5 point. Each of those who have lost receives 0 point.

Round Two

8.1

The players are arranged in groups of the same score.

8.2

If the number of players is uneven, then the pairing-allocated bye is awarded as in Rule 1.

8.3

Pairing begins with the highest score-group (1 point), continues with the lowest score-group (0 point) and finishes with the Median Score-group (0.5 point).
Detailed instructions for pairing Round Two and subsequent rounds are above.