The main tie-break of the Swiss system, explained end to end: all five formula variants (BH, BH-C1, BH-C2, BH-M1 and SB for comparison), two worked examples with every number spelled out, the FIDE Handbook C.07 edition effective 1 March 2026, the capped dummy opponent for byes and forfeits, and the famous cases where the tie-break decided a title. Written for arbiters, organizers and the player who wants to know why they finished second.
The Buchholz score is the sum of the final scores of all of a player's opponents in a Swiss tournament. The better your opponents did, the higher your Buchholz. It is the standard way of ordering players who finished on the same number of points, and it is governed by FIDE Handbook C.07, «Play-Off and Tie-Break Regulations», in the edition effective from 1 March 2026.
The reasoning is straightforward. If two players both score 4½ out of 6, but one of them spent the event against the leaders while the other was paired down into the tail of the field, it is fairer to put the first one higher. Buchholz turns that intuition into a single number.
It belongs to the Swiss system - opens, club leagues, scholastic events, Lichess Swiss, Chess.com Swiss. It does not work in a round-robin, where everyone faces the same field and the sums come out the same for everybody; there you use the Sonneborn-Berger score, and the round schedule comes from the Berger tables.
Bruno Buchholz was secretary and tournament director of the 25th Congress of the German Chess Federation in Magdeburg in 1927. Little is known about his life - no birth date survives and he is thought to have died around 1958. His lasting mark on chess is the ranking system he proposed for Swiss events.
The Czech master Oskar Gelbfuhs proposed a way of ranking players tied on points in a round-robin by weighting wins according to the strength of the opposition. This is the ancestor of the Sonneborn-Berger score.
William Sonneborn and Johann Berger were the first to apply the method in practice, at a round-robin in Liverpool. The system still carries their names and FIDE still uses it as the primary tie-break for round-robin events.
Bruno Buchholz, a German player and organizer from Magdeburg, proposed a new system specifically for Swiss tournaments and first used it at an event in Bitterfeld where he was himself a competitor. He originally computed a «weight» - the sum of the opponents' scores multiplied by the player's own score. The modern version, a plain sum, came later.
As the Swiss system took over mass open tournaments, Buchholz became the default ranking tool for players tied on points. FIDE codified it in Handbook C.07.
FIDE reworked how missed rounds are counted. From September 2023 a pairing-allocated bye and a forfeit win were treated as a game against a «virtual opponent» whose score equalled the player's own. That solved the old problem of a bye wrecking your Buchholz.
FIDE approved an expanded tie-break regulation effective 1 March 2026: the dummy opponent's score is now capped, which pushes tie-break numbers down; Standard Points were introduced for events using unusual scoring; starting number and rating were added so that events reach a drawing of lots less often; and using Buchholz in a round-robin is now explicitly ruled out.
In short: BH is the plain sum; BH-C1 drops the weakest opponent; BH-C2 drops the two weakest; BH-M1 drops the weakest and the strongest; SB is the weighted sum, shown here for contrast. The notation follows C.07 as effective from 1 March 2026.
BH(p) = Σ S(o_i)
The sum of the final scores of all the player's opponents. The simplest variant and historically the original one.
Where it is used: The base figure, usually applied as a secondary tie-break after BH-C1.
BH-C1(p) = Σ S(o_i) − min{S(o_i)}The sum with the single weakest opponent removed. It cancels out the unfairness of being handed one very weak opponent by the pairing. FIDE Handbook C.07, Article 14.1.1.
Where it is used: The main FIDE tie-break for the Swiss system.
BH-C2(p) = Σ S(o_i) − [min₁ + min₂]
The sum with the two weakest opponents removed. Used in long Swiss events of nine rounds or more, where the spread in opponent strength is wider.
Where it is used: Large opens and championships with 9+ rounds.
BH-M1(p) = Σ S(o_i) − min − max
The sum with both the weakest and the strongest opponent removed. Long known in the United States as the Harkness median.
Where it is used: USCF events and scholastic Swisses in the US and Canada.
SB(p) = Σ (S(o_i) × R(p, o_i))
The opponents' scores weighted by your result against them: beat an opponent and you take their whole score, draw and you take half, lose and you take nothing.
Where it is used: The main FIDE tie-break for round-robins, a secondary one for Swiss events.
A modelled Swiss event. The winner F (4½) needed no tie-break. Only one player, A, finished on 4. Two players, B and E, finished on 3, and that is where the arithmetic gets interesting.
| Player | Score | Opponents by round (their final scores) |
|---|---|---|
| A (wins the tie-break among players level on points) | 4 | B (3), F (4½), D (2), E (3), G (2) |
| B | 3 | A (4), bye, C (1½), F (4½), G (2) |
| E | 3 | C (1½), F (4½), G (2), A (4), bye |
A faced B (3), F (4½), D (2), E (3) and G (2). Add up those five final scores.
Both had one bye. Under the current regulations the missed round counts as a game against a dummy opponent whose score equals the player's own final score, but no higher than the cap: the points for a draw multiplied by the number of rounds. In a five-round event that is ½ × 5 = 2½. Both players finished on 3, so their dummy is capped at 2½ rather than 3.
This is exactly why regulations list several tie-breaks in sequence. Buchholz comes first, but when it ties you move to SB, then to the direct encounter, then to rating performance, and so on down the list you published before round one.
A clean example with no dummy opponents anywhere. This one shows Buchholz separating several ties at once - and shows its most misunderstood property.
Each cell shows the opponent and the result: 1 for a win, ½ for a draw, 0 for a loss.
| Player | R1 | R2 | R3 | R4 | R5 | Score |
|---|---|---|---|---|---|---|
| Anna | Dina 1 | Boris ½ | Vera 1 | Gleb ½ | Zhanna 1 | 4 |
| Boris | Egor 1 | Anna ½ | Gleb 1 | Vera 1 | Zurab ½ | 4 |
| Vera | Zhanna ½ | Gleb 1 | Anna 0 | Boris 0 | Dina 1 | 2½ |
| Gleb | Zurab 1 | Vera 0 | Boris 0 | Anna ½ | Egor 1 | 2½ |
| Dina | Anna 0 | Zhanna 1 | Egor ½ | Zurab 1 | Vera 0 | 2½ |
| Egor | Boris 0 | Zurab 1 | Dina ½ | Zhanna ½ | Gleb 0 | 2 |
| Zhanna | Vera ½ | Dina 0 | Zurab 1 | Egor ½ | Anna 0 | 2 |
| Zurab | Gleb 0 | Egor 0 | Zhanna 0 | Dina 0 | Boris ½ | ½ |
Points decide first; on equal points compare Buchholz (BH), then Buchholz Cut-1 (BH-C1), then Sonneborn-Berger (SB).
| Place | Player | Score | BH | BH-C1 | SB |
|---|---|---|---|---|---|
| 1 | Anna | 4 | 13½ | 11½ | 10¼ |
| 2 | Boris | 4 | 11½ | 11 | 9¼ |
| 3 | Vera | 2½ | 15 | 13 | 6 |
| 4 | Gleb | 2½ | 13 | 12½ | 4½ |
| 5 | Dina | 2½ | 11 | 10½ | 3½ |
| 6 | Egor | 2 | 11½ | 11 | 2¾ |
| 7 | Zhanna | 2 | 11½ | 11 | 2¾ |
| 8 | Zurab | ½ | 13 | 11 | 2 |
Bar length is the Buchholz score, players in finishing order. Within a score group it sets the priority, but across the table it does not decrease monotonically: last-placed Zurab (½) has a bigger Buchholz than runner-up Boris.
Buchholz is defined on the final scores of the whole field, so it can only be computed after the last round. Anything calculated earlier will move.
For the player whose tie-break you are computing, write out every opponent round by round, along with the result of each game.
Take the final crosstable and note what each of those opponents finished on: 1, 1½, 2, 2½ and so on.
For a bye or a win by forfeit, add a dummy opponent carrying the player's own final score, but no higher than the cap: half the tournament maximum for a voluntary unplayed round, or the scheduled opponent's adjusted score for a forfeit. The cap was introduced in the edition effective 1 March 2026.
That total is the plain Buchholz, BH. Every Cut and Median variant is derived from it.
For BH-C1 subtract the smallest opponent score. For BH-C2 subtract the two smallest. Note that where the player has an unplayed round, the regulations require the cut to remove that contribution first, regardless of whether it is numerically the smallest.
Compare the result only against players on the same number of points. The higher Buchholz finishes higher. Comparing across score groups is meaningless.
The loudest case in recent memory. After 13 rounds four players shared first on 9½: Nodirbek Abdusattorov, Ian Nepomniachtchi, Magnus Carlsen and Fabiano Caruana. The regulations sent the two with the highest Buchholz Cut-1 into the play-off, and those were Abdusattorov and Nepomniachtchi. The reigning world champion Carlsen and Caruana were eliminated on tie-break alone, which set off a long public argument about whether cutting a field that way is fair. Abdusattorov won the play-off and became the youngest world rapid champion in history at 17.
The Moscow open traditionally produces a large pile-up at the top, and Buchholz has decided first place there repeatedly: Gregory Kaidanov in 2002 out of a five-way tie, Emil Sutovsky in 2005, Etienne Bacrot in 2009 ahead of Alexander Moiseenko, Evgeny Najer in 2016. In a Swiss with a big field, sharing first is the norm rather than the exception.
Team events use «Olympiad Sonneborn-Berger», the team-level analogue: the sum of the opponents' match points weighted by the match result against them, with the weakest match dropped. India won both golds convincingly in Budapest 2024, but in most previous Olympiads the silver and bronze were settled precisely on this coefficient.
The recurring mistakes arbiters and organizers make when computing Buchholz by hand. Every item traces back to FIDE Handbook C.07 and Arbiters' Commission material.
Wrong. You take the opponent's final score at the end of the whole event, not what they had when you played them.
The pre-2023 mistake: writing 0 into the tie-break for a bye. Current rules substitute a dummy opponent carrying the player's own score, capped since March 2026.
Same dummy-opponent rule. A win by forfeit counts as a game against a dummy, with the dummy's score capped at the scheduled opponent's adjusted score.
An opponent's own unplayed rounds are evaluated under the unplayed-rounds article, normally as draws - not as zeros.
Where a player has an unplayed round, the regulations require the cut to remove that contribution first, even when it is not numerically the smallest.
Median trims from both ends, worst and best. Cut trims only from the bottom. The two give different numbers and are not interchangeable.
Methodologically wrong: the score is only defined once the field's final scores are known. Interim figures mislead players about where they stand.
A withdrawn player's remaining rounds are treated as unplayed, which materially changes the Buchholz of everyone who faced them.
Vibe Chess is a free tournament service in the browser and through a Telegram bot. It computes every Buchholz variant under FIDE Handbook C.07 - BH, BH-C1, BH-C2, BH-M1 - plus Sonneborn-Berger, and it handles byes and forfeits through the dummy-opponent rule so you do not have to. Suitable for schools, corporate events, club leagues and casual evenings.
It is the sum of the final scores of all your opponents. The better your opponents did across the tournament, the higher your Buchholz. It is used in the Swiss system to order players who finished on the same number of points. The system was invented by the German player Bruno Buchholz in 1932.
BH (sometimes «Bch» or just «Buch») is the Buchholz score. Next to it you will usually find BH-C1, which is Buchholz with the weakest opponent cut, and SB, the Sonneborn-Berger score. The ordering column in most crosstables is BH-C1 rather than plain BH.
FIDE Handbook C.07 recommends BH-C1 as the primary Swiss tie-break: dropping the weakest opponent removes the luck of the pairing, where one player happens to be handed a much weaker opponent than the other. Plain BH comes second.
Buchholz is just the sum of your opponents' scores, regardless of how you did against them. Sonneborn-Berger is a weighted sum: beat an opponent and you take their whole score, draw and you take half, lose and you take nothing. SB is the primary tie-break in round-robins, Buchholz in Swiss events.
Bruno Buchholz, a German player and organizer from Magdeburg, first used it at a tournament in Bitterfeld in 1932. He called it an auxiliary points scale rather than a tie-break, and his original version multiplied the sum of the opponents' scores by the player's own score.
No, and since March 2026 the regulations say so explicitly. In a round-robin everybody plays everybody, so the sum of the opponents' scores is nearly identical for all players and the tie-break carries no information. Round-robins use Sonneborn-Berger, the direct encounter and the Koya system instead.
The missed round counts as a game against a dummy opponent, with the result matching the points awarded, so a full-point bye counts as a win. The dummy's score equals the player's own final score, but since 1 March 2026 it is capped: for a voluntary unplayed round the cap is the points for a draw multiplied by the number of rounds, which is half the tournament maximum; for forfeits it is the scheduled opponent's adjusted score.
Their real score at the point of withdrawal is fixed, and their remaining unplayed rounds are evaluated under the same unplayed-rounds article, normally as draws, when you compute the Buchholz of everyone who played them.
Only after the final round, because it depends on the final scores of the whole field. Interim Buchholz figures published mid-event on sites like chess-results.com are informational only and routinely change in the last round.
It is FIDE's device for rounds where a player has no game: a pairing-allocated bye, a win by forfeit, a voluntary half-point bye. Without it, a bye would contribute nothing and wreck the player's tie-break. The dummy is given the player's own final score, capped since March 2026 at half the tournament maximum for voluntary unplayed rounds and at the scheduled opponent's adjusted score for forfeits. The same capped figure feeds both Buchholz and Sonneborn-Berger.
There is no single mandated sequence - the tournament regulations decide, and FIDE publishes recommended sequences rather than one default. A common heterogeneous-field sequence is BH-C1, then BH, then Sonneborn-Berger, then the direct encounter and number of wins. Whatever you use, publish it before round one.
Sonneborn-Berger only, computed under the FIDE rule with a virtual opponent for unplayed games. One number, one algorithm, no regulation settings to configure.
Eight levels in sequence: Buchholz Cut-1, Buchholz, Sonneborn-Berger, direct encounter, number of wins, wins with Black, AROC-1 and rating. It is the most elaborate list among the major online platforms.
Yes, famously. At the 2021 World Rapid Championship in Warsaw four players shared first after 13 rounds: Abdusattorov, Nepomniachtchi, Carlsen and Caruana. The regulations sent the two with the best Buchholz Cut-1 to the play-off, which meant Carlsen and Caruana were eliminated on tie-break. Abdusattorov won the play-off and became the youngest world rapid champion at 17.
Yes. Take the final crosstable, write down the final score of each of the player's opponents and add them up. For BH-C1 subtract the smallest of those. For SB, multiply each opponent's score by your result against them - 1, ½ or 0 - before adding.
Standard Points were introduced so that events with alternative scoring, such as 3-1-0, can be normalised back to the classical 1/½/0. The list of tie-breaks was extended. The treatment of unplayed games in Buchholz and Sonneborn-Berger was rewritten in more detail, including the cap on the dummy opponent. And the regulations lean harder on rating and starting number so that events reach a drawing of lots less often.
The full organizer walkthrough is in how to organize a chess tournament. For the round-robin tie-break and a working calculator, see the Sonneborn-Berger calculator, and for round-robin schedules the Berger tables. On how long a Swiss needs to be, see how many rounds a Swiss system needs, and the general questions are collected in the tournament FAQ.