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Buchholz tiebreak explained

The sum of your opponents’ scores — and the surprisingly awkward question of what to do about the games nobody played.

Last updated: 25 July 2026 · Rules verified against the current FIDE and US Chess texts on 25 July 2026.

The tiebreak Carlsen called “completely idiotic”

At the 2021 World Rapid Championship four players finished on 9½/13. Two went through to a playoff for the world title and two went home, and the rule that decided which was which was Buchholz Cut 1. It sent Nodirbek Abdusattorov and Ian Nepomniachtchi to the playoff and eliminated Magnus Carlsen and Fabiano Caruana. Carlsen’s verdict: a “completely idiotic rule”.

He was half right. Buchholz does not measure how well you played; it measures how hard your schedule was, and it moves on boards you never sat at. But every tiebreak is a proxy, and somebody has to be handed the trophy.

What Buchholz actually measures

The definition is one line. FIDE Article 8.1: Buchholz is “the sum of the scores of each of the opponents of a participant”. Higher is better, because it means you played tougher people. Two errors are common online, including on some very large chess sites:

• You do not multiply by your own score. Weighting by your own result is a different tiebreak (Sonneborn-Berger).
• You use opponents’ final scores, not their scores when you played them. Somebody you beat in round one who then wins four straight helps you a lot.

One prohibition is worth memorising because it is short and almost never quoted: the note to Article 8 says Buchholz must not be used in round robins. In an all-play-all everybody faces the same field, so the sums are nearly identical and what differences remain are an artefact. Use Sonneborn-Berger for a round robin instead.

The Buchholz family: Cut-1, Cut-2, Median-1, Median-2

Plain Buchholz has an obvious weakness: one opponent who collapses or withdraws drags your total down through no fault of yours. The modifiers throw outliers away before summing.

Tiebreak calculator

Enter one player's tournament: the final score of each opponent they faced, and the result against them. Every tiebreak below is recalculated as you type. A bye or an unplayed game counts for your score but is excluded from the tiebreaks, which is what both FIDE and Chess Caddy do.

One crosstable, three different winners

Here is why the modifier is not a detail. Three players tie for first on 4/5 in a 14-player Swiss. These are the final scores of the five opponents each of them faced.

Check it. Ana: 4+3+3+2+1 = 13; drop the 1 for 12; drop the 1 and the 4 for 8. Bo: 4+3+2½+2+2 = 13½; drop a 2 for 11½; drop a 2 and the 4 for 7½. Cai: 3+3+3+2½+1 = 12½; drop the 1 for 11½; drop the 1 and a 3 for 8½. Plain Buchholz gives it to Bo, Cut-1 to Ana, Median-1 to Cai — same crosstable, three champions.

The reason is in the numbers. Bo faced the toughest field but had no weak link to discard, so cutting helps Bo least. Ana had one opponent who fell apart, and Cut-1 forgives exactly that. Cai never played anybody above 3 — the flattest schedule — which is the shape Median-1 rewards, because stripping both extremes leaves that solid middle intact. Publish your tiebreak order before round one: afterwards, every choice looks like it was made to favour somebody.

Sonneborn-Berger, and why round robins use it

Sonneborn-Berger (Article 9.1) asks a better question: not just who you played but how you did against them. For each round multiply the opponent’s final score by the points you scored against them — full for a win, half for a draw, nothing for a loss. A five-player round robin:

Ana and Bo tie on 3/4 and their own game was drawn, so head-to-head settles nothing. Sonneborn-Berger does:

• Ana = (½×3) + (1×2½) + (½×1½) + (1×0) = 1½ + 2½ + ¾ = 4.75
• Bo = (½×3) + (½×2½) + (1×1½) + (1×0) = 1½ + 1¼ + 1½ = 4.25

Ana wins, for exactly the right reason: Ana beat the third-placed player, while Bo only drew with them and made up the difference against the tail. Beating Eli, on zero, was worth nothing to either.

Sonneborn-Berger is the round-robin standard because everyone faces the same opponents, so the only thing left to separate two tied players is which games they won. In a Swiss, where opponent sets genuinely differ, Buchholz asks the better question.

Cumulative, direct encounter and the rest

Cumulative (progressive, Article 7.5) adds up your running score after each round. Two players finish on 4/5: one won, won, won, lost, won and scores 1 + 2 + 3 + 3 + 4 = 13; the other lost then won four straight and scores 0 + 1 + 2 + 3 + 4 = 10. It rewards leading early, since leaders face the hardest available opponents, and needs no information about anybody else.

Direct encounter is the most intuitive tiebreak and the least usable: if the tied players all met, their results against each other decide. With three tied in a five-round Swiss they usually did not all meet, so it does not apply.

The current FIDE chapter lists a long menu. The ones worth knowing:

One gating rule matters for club software: under Article 10, rating-based tiebreaks such as ARO must be excluded when unrated players are in the field — which quietly rules them out of most club events.

The bye problem, and FIDE’s 2026 dummy opponent

Here is the part that breaks every simple explanation. Buchholz is the sum of your opponents’ scores — so what do you add for a round with no opponent? Not nothing: if unplayed rounds counted zero, anyone taking a half-point bye would be crushed on tiebreak for missing a round they were entitled to miss. FIDE’s answer is to invent an opponent.

What you contribute to other people (Article 16.3)

The easy direction first. When your opponents add up their Buchholz, your score goes in at face value — including points from forfeit wins and from a pairing-allocated bye. One exception: a requested bye not followed by a real game, which includes a bye in the last round, is downgraded so that round contributes only ½. Somebody who books off and never returns should not inflate the tiebreaks of everyone they beat earlier.

What gets added for your own unplayed round (Article 16.4)

For your own unplayed rounds FIDE substitutes a dummy opponent whose score is your own final score, on the reasoning that you would on average have been paired against somebody at your own level. Then come two caps, the second being the March 2026 change:

• 16.4.1 — forfeits. Capped at the scheduled opponent’s adjusted score. If the player who never turned up finished on 1½, your dummy is worth 1½, not your own score.
• 16.4.2 — byes. Capped at draw points times the number of rounds — that is, 50% of the tournament.

Work the cap through. In a five-round event the cap is ½ × 5 = 2½. A player finishes on 4½/5, having won four games and taken a half-point bye in round three. Their four real opponents finished on 3½, 3, 2½ and 2 — a sum of 11.

Two whole points of difference, on a tiebreak where placings turn on half a point. Do not trust a worked example unless it is dated after March 2026 — including FIDE’s own exercise set, written for the older text.

The counterintuitive bit (Article 16.5)

Now apply Cut-1 to that player. The five contributions are 3½, 3, 2½, 2 and a dummy worth 2½. The obvious move is to discard the lowest, the 2, giving 11½.

That is wrong. Article 16.5 says the cut must discard the bye’s contribution first, even when the bye is not the numerically lowest item. Cut-1 therefore discards the dummy and the answer is 11, not 11½.

“Dummy opponent” is not “virtual opponent”

These two terms are used interchangeably across the internet and are not the same thing. FIDE’s own arbiter material says so.

• Dummy opponent is the current tiebreak construct above: the imaginary player substituted for your unplayed round, capped under Article 16.4.
• Virtual opponent is two other things: a pairing concept in the Swiss chapter C.04, used in bracket arithmetic; and the name of the superseded tiebreak rule applying before September 2023, which built a running notional score from your score when paired, the result, and half a point per remaining round.

If a page or a piece of software says it computes Buchholz using the “virtual opponent”, it is describing a rule FIDE retired years ago and revised again in 2026.

FIDE vs US Chess: opposite philosophies

US Chess handles unplayed games in almost exactly the reverse way. That is why Swiss-Manager and WinTD can produce different tiebreak numbers from the same crosstable and both be right.

The terminology differs too:

• Solkoff = Buchholz. Plain sum of opponents’ scores.
• Median (or Harkness) = Median-1. Drop the highest and lowest.
• Modified Median has no FIDE equivalent. What you drop depends on your own score: a plus score drops the lowest, a minus score drops the highest, an even score drops both. A leader should not be punished by one collapsed opponent, and a player on a minus score should not be flattered by having been paired up early.

US Chess, unlike FIDE, publishes a default order: Modified Median → Solkoff → Cumulative → Cumulative of Opposition.

There is no FIDE default tiebreak order

So many pages get this wrong that it is worth stating plainly. FIDE defines no default tiebreak sequence. Article 4.1 delegates the ordered list to the Chief Organiser, who must publish it before the first round.

There is therefore no such thing as “the FIDE default tiebreak”. Buchholz Cut-1 then Buchholz is what large FIDE Swiss events conventionally use — a convention, not a rule. Your job is to decide, write it into the tournament announcement, and not change it afterwards.

Which tiebreak should you actually pick?

Prizes: split the cash, tiebreak the trophy

One convention removes most tiebreak arguments before they start: cash prizes are pooled and split evenly among tied players, and tiebreaks decide only what you cannot cut in half — trophies, titles, board-one seeding, qualification places.

Two players tie for first with £200 and £100 on offer? Pool the £300 and pay £150 each. Nobody then has to accept that their year was decided on board nine.

What Chess Caddy does — and doesn’t

So as not to add to the pile of vague claims: Chess Caddy offers exactly three tiebreaks — Buchholz, Sonneborn-Berger and Cumulative. No Cut-1, no Median, no Modified Median, no head-to-head, no rating-based tiebreak. If you need Cut-1, you need different software.

More on byes, forfeits and withdrawals on the FAQ page.

Frequently asked questions