What is your Fischer number?

by Frederic Friedel
8/13/2020 – How far away are you from the World Champion, in terms of handshakes, we asked our readers? At least one traced his handshake route all the way back to Philidor. Now we have a new challenge: have you beaten someone who has beaten someone who has beaten a World Champion? Prof. Christian Hesse describes the idea. You are invited to participate.

Master Class Vol.1: Bobby Fischer Master Class Vol.1: Bobby Fischer

No other World Champion was more infamous both inside and outside the chess world than Bobby Fischer. On this DVD, a team of experts shows you the winning techniques and strategies employed by the 11th World Champion.

Grandmaster Dorian Rogozenco delves into Fischer’s openings, and retraces the development of his repertoire. What variations did Fischer play, and what sources did he use to arm himself against the best Soviet players? Mihail Marin explains Fischer’s particular style and his special strategic talent in annotated games against Spassky, Taimanov and other greats. Karsten Müller is not just a leading international endgame expert, but also a true Fischer connoisseur.


The following article appeared in The Joys of Chess by Christian Hesse. We reproduce a chapter from his book (2011, pp.95-98), in excerpts with kind permission of the author.

The conqueror of the conqueror of Fischer

At the end of the 60s the American psychologist S. Milgram conducted an experiment which has since become famous. He was investigating the degree of linking up in social networks.

A total of 96 Americans was chosen, completely at random, and Milgram gave to each of them a letter and the description of a final person (i.e. name, where he or she lived, job – specifically it was a stockbroker in Massachusetts), for whom the letter was intended. The participants were each asked to send the letter to an acquaintance of theirs whom they believed would be closer to that final person. The acquaintance also received the same instructions.

Milgram established that the average length of each chain of people consisting of pairwise acquaintances was six persons. It was called the "small world phenomenon" and the familiar concept of the six degrees of separation was minted.

Just recently the sociologists P. Dodds and D. Watts and colleagues repeated this experiment on an international level. They managed to recruit for it more than 60,000 participants in 116 countries. Each participant was given the name of a final person (one of a total of 18 people in the world), but instead of letters communication was by e-mail over the Internet this time.

The results essentially confirm Milgram’s findings. It seems to be the case that any two people on the planet can be linked by a chain of only a few mutually acquainted people. This is an indication of the strength of the interrelationship of humanity by personal acquaintanceship.

This brings us to the real subject of this chapter. It is instructive to take a look at the international community of chess players from this angle. What interests us is not collaboration but rather the competitive aspect.

Let us first introduce a new number: the Fischer number. In the opinion of many people Bobby Fischer was the greatest player of all time. (Tal: “Fischer is the greatest genius who ever descended from chess heaven to earth.” Kasparov: “I consider Fischer to be the greatest world champion”). A victory over Fischer in a serious game was always a special event. But few people managed to bring about this result.

But the group of those who defeated a player who had beaten Fischer at least once is bigger. As in turn is the group consisting of those players who had conquered a player who had beaten a player who had defeated Fischer. Based on this principle of iteration, we then construct for every chess player a so-called Fischer number. This is defined in the following concrete terms:

  • Bobby Fischer himself has the Fischer number 0.
  • Anyone who defeated Fischer at least once gets the Fischer number 1.
  • Anyone (apart from Fischer) who beat at least once someone with the Fischer number 1 gets the Fischer number 2, etc.
  • Anyone who cannot point to a chain of victories leading back to Fischer has the Fischer number ∞ (infinity).

The smaller the Fischer number, the greater the performance. If your Fischer number is n, then you are in a certain way n-1 steps away from a possible victory over the greatest player of all times.

Of course the Fischer number is not designed to be a strict measurement of chess performance in the same way as the Elo number, but it should be understood rather as a contribution to chess folklore. You work out your Fischer number by setting out the shortest possible unbroken line of victories between you and Fischer. Even if you are only a casual player: Who is the strongest player you have ever defeated? Perhaps that player once beat a club player, and the latter once defeated a local champion, and the local champion defeated a regional champion, who beat someone playing in the Bundesliga. And the Bundesliga player beat an international master, and this international master defeated a grandmaster and the grandmaster in turn beat Korchnoi. Korchnoi has the Fischer number 1. Then your own Fischer number would be no bigger than 9.

Some other Fischer numbers which can easily be derived from some database research are: Spassky = 1, Reshevsky = 1, Tal = 1, Geller = 1, Karpov = 2, Kasparov = 2, Deep Blue = 3.

Your author’s Fischer number is in fact 6 and is based on the following pathway through the chess graph:

  • Boris Spassky defeated Fischer (several times)
  • Harald Lieb defeated Spassky (Munich 1979)
  • Werner Nautsch defeated Lieb (Bundesliga 1981)
  • Frank Beckmann defeated Nautsch (NRW-Liga 1992)
  • Bernd Sakulski defeated Beckmann (Plettenberg 1986)
  • Christian Hesse defeated Sakulski (Attendorn Championship 1980/81)

In this chapter my main aim is to introduce the idea of Fischer numbers and if possible encourage the study of their characteristics. There are many interesting questions which surround them. I should like to advance my conjecture that the vast majority of chess players all the way down to club players, have finite Fischer numbers, i.e. they can put together a succession of won games which leads back to a victory over Fischer. It can further be supposed that the graph of all chess players with finite Fischer numbers (the Fischer graph) exhibits the small world property and that the average Fischer number in this Fischer graph is no greater than perhaps 6 or 7.

As well as the investigation of this hypothesis there is another interesting point: of course, just like Fischer numbers we can introduce Kramnik numbers, and in fact for any chess player one wants. How do Fischer numbers behave compared to Kramnik numbers, as far as the average is concerned and regarding the characteristics of their distribution? Is the chess community on average further away from a victory over Fischer than from a victory over Kramnik? What is the largest known finite Fischer number or Kramnik number belonging to a titled player?

Christian Hesse holds a Ph.D. from Harvard University and was on the faculty of the University of California at Berkeley until 1991. Since then he is Professor of Mathematics at the University of Stuttgart (Germany).

Subsequently he has been a visiting researcher and invited lecturer at universities around the world, ranging from the Australian National University, Canberra, to the University of Concepcion, Chile. Christian Hesse is one of the most prominent living German mathematicians.

In 2007 he authored “Expeditionen in die Schachwelt” (Expeditions into the world of chess), a collection of about 100 essays that the Viennese newspaper Der Standard called “one of the most intellectually scintillating and recommendable books on chess ever written.”

In 2011 New in Chess published an English language version of the book.

The Joys of Chess is an unforgettable intellectual expedition to the remotest corners of the Royal Game. En route, intriguing thought experiments, strange insights and hilarious jokes will offer vistas you have never seen before.

Christian Hesse is a Harvard-trained professor of Mathematics who has taught at the University of California, Berkeley (USA), and since 1991 at the University of Stuttgart. Chess and literature are his main hobbies, and he also likes fitness and boxing. His heroes are the ones who fall to the bottom and rise again, fall and rise again…

Tell us your Winchain number

By Frederic Friedel

My own winchain number to World Champions could be construed as 2 – I defeated the fledgling Fritz 1 (in 1991) a number of time, and in fact won a couple games against Fritz 2. Both Kasparov and Kramnik have lost games to Fritz. But that does not count: first of all I did not beat the same program as the one that beat the World Champions; and secondly, my games were not under any formal conditions. In addition I believe we should not include computer programs in the winchain.

Well, maybe I am just two steps away from a Vice Champion?! In 1981 a lad named Martin was staying with us, and I played him (blindfold) in an informal game, which I won – to the abject horror of his brother ("I am so ashamed – my brother loses to a patzer!"). The picture is of Martin during the game.

Now Martin is the older brother of Nigel Short, and learnt the game before him. I am sure that he beat the toddler in the beginning, when he had just learnt the rules. So I beat someone who beat a (future) Vice Champion.

But again this does not count: it was an informal game, not recorded anywhere. For our winchains we will insist on formal encounters, in tournaments or opens. I suggest that simuls do not count, or only conditionally. Put a "(Sim)" behind the name of a player you have beaten in a simultaneous exhibition.

Well, I am eager to see what kind of winchains our readers will find. Tell us in the feedback section below. In few days I will show you a very nice little ChessBase utility (some readers will know it) that can help tremendously in the search.

Editor-in-Chief emeritus of the ChessBase News page. Studied Philosophy and Linguistics at the University of Hamburg and Oxford, graduating with a thesis on speech act theory and moral language. He started a university career but switched to science journalism, producing documentaries for German TV. In 1986 he co-founded ChessBase.


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