Buchholz: Chess tie-break system

Buchholz

Definition

Buchholz is a tie-break system used primarily in Swiss-system tournaments. It is defined as the sum of the final scores of a player’s opponents. In other words, your Buchholz score measures how strongly your opposition performed overall, independent of your own game results against them.

Synonyms you may encounter include “Sum of Opponents’ Scores” (SOS). Related but different systems include Sonneborn–Berger (which weights the opponents’ scores by your result) and Median-Buchholz (which trims extreme values).

How it is used

When two or more players finish on the same number of points, organizers apply pre-announced tie-breaks to rank them. Buchholz is one of the most common first or second tie-breaks in open Swiss events, scholastic tournaments, and many online Swiss competitions. It rewards players who faced opponents that scored well across the event, reflecting the “toughness” of a player’s path.

Typical tie-break orders may include some combination of: Direct Encounter, Buchholz Cut 1, Buchholz, Median-Buchholz, Sonneborn–Berger, and rating-based averages. The exact order varies by event—always check the regulations.

Calculation (step by step)

• List all opponents you actually played.
• Record each opponent’s final score at the end of the tournament.
• Sum those scores; the total is your Buchholz.

Example (5-round Swiss): You score 4/5 against opponents whose final scores are 3.0, 2.5, 4.0, 4.5, and 3.5. Your Buchholz is 3.0 + 2.5 + 4.0 + 4.5 + 3.5 = 17.5.

Event regulations specify how to treat special cases such as byes, forfeits, or opponents with unplayed games. Many tournaments mitigate anomalies by using a “cut” or “median” version (see below).

Common variants

• Buchholz Cut 1 (or “Cut-1”): Sum of opponents’ scores after removing the single lowest opponent score. This helps neutralize distortions from facing one very low-scoring opponent or receiving a bye.
• Buchholz Cut 2: Remove the two lowest opponent scores before summing.
• Median-Buchholz: Remove the single highest and the single lowest opponent scores; sum the rest. (For short events, organizers sometimes adjust the trimming.)
• Average Buchholz: The Buchholz total divided by the number of rounds; used less frequently but normalizes for different event lengths.
• Team Swiss adaptation: In team events, a “Buchholz (match points)” version sums the total match points scored by a team’s opponents.

Strategic significance

Since Buchholz reflects your opponents’ final performance, you effectively “root” for your previous opponents to continue scoring well—they lift your tie-breaks. Practically, players can’t directly manipulate Buchholz, but understanding it helps:

• Beating or drawing players who end up near the top often boosts your Buchholz significantly.
• Early pairings against strong or in-form opposition usually lead to healthier Buchholz later.
• In the final round, a quick draw might be risky if your Buchholz is weak relative to rivals; sometimes a win against a high-scoring opponent is the only way to overcome inferior tie-breaks.

Worked examples

Example A (5 rounds): Player A scores 4.0/5. Opponents’ final scores: 3.0, 2.5, 4.0, 4.5, 3.5.

• Buchholz = 3.0 + 2.5 + 4.0 + 4.5 + 3.5 = 17.5
• Buchholz Cut 1 (drop lowest 2.5) = 15.0
• Median-Buchholz (drop highest 4.5 and lowest 2.5) = 3.0 + 4.0 + 3.5 = 10.5
• Average Buchholz = 17.5 / 5 = 3.5

Example B (comparison): Player B also scores 4.0/5. Opponents’ final scores: 2.0, 3.0, 3.5, 3.5, 4.0.

• Buchholz = 2.0 + 3.0 + 3.5 + 3.5 + 4.0 = 16.0
• Buchholz Cut 1 (drop lowest 2.0) = 14.0

Despite equal points, Player A outranks Player B on both Buchholz (17.5 vs 16.0) and Cut 1 (15.0 vs 14.0), reflecting tougher opposition.

Historical and practical notes

• The system is named after the German mathematician Bruno Buchholz and became widely adopted with the growth of Swiss-paired tournaments in the 20th century.
• Many national federations and international events list Buchholz (often with a cut) among primary tie-breaks for Swiss events. Playoffs are sometimes used for the title, with Buchholz settling other prizes or seeding.
• Compared with Sonneborn–Berger, Buchholz is simpler: it doesn’t weight by your result; it simply aggregates how your opponents finished.
• Debate occasionally arises about fairness when byes/forfeits occur; “cut” and “median” variants are popular precisely because they reduce noise from such anomalies.

Common pitfalls and clarifications

• Buchholz is not a performance rating; it doesn’t use ratings at all—only opponents’ final scores.
• Your own score is not added into your Buchholz; only your opponents’ scores count.
• Winning by a larger margin in a game doesn’t matter; only the final standings of your opponents influence Buchholz.
• Always check the event’s tie-break order and the exact definitions (e.g., how byes and forfeits are treated), as implementation details can vary.

At-a-glance summary

• What it measures: The strength of your opposition via their final scores.
• Why it matters: Breaks ties among players with equal points in Swiss events.
• Key variants: Cut 1, Cut 2, Median-Buchholz, Average Buchholz.
• Player takeaway: Stronger and successful opposition boosts your tie-breaks—cheer for your former opponents!