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.

Topics: Bobby Fischer

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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Hammerchewer Hammerchewer 8/25/2020 05:52
I am sure that many senior players must have a Fischer number of 3 by virtue of having beaten somebody (even if only in a simul) who in turn beat Spassky (or another GM of that generation) who of course beat Fischer several times though he lost their matches. Anyone who beat Spassky himself has a Fischer number of 2.
In my case I beat (in a real tournament game) an English schoolboy (already rated higher than me at the time) who went on to win at least one game against every postwar world champion (bar Fischer and Botvinnik whom he was too young to meet), albeit in a couple of cases it was in a blitz or rapid tournament. So my Fischer and Botvinnik numbers are 3 and my number for the other champions is 2.
Working out chains to older champions is more challenging.
I think HollyHampstead's Morphy winchain number of 4 is impressive. My only game with Scottish international Dr Aitken was drawn and he was not that weak that "zillions" of living people have beaten him. My peak rating was over 2200 and I have several ways of reaching a Morphy number of 4 if results don't matter but I haven't yet found a way to do better than 5 if every link in the chain must be a win.
The main problem is that so few players with long careers beat Morphy. The chain has to go through Paulsen or the Rev. John Owen or Bird; the latter claimed to have won a game against Morphy though it was not preserved. I tried to find a route involving George Alan Thomas as my number 2; he came to England about 1896 but if he ever played with Bird or Owen it is not recorded.
A complete list of people in Britain who beat Mieses after WW2 would be useful...
HollyHampstead HollyHampstead 8/24/2020 02:46
My Morphy number is 4. I believe that there are zillions of other players who have a Morphy number of 4. (I beat Dr J. M. Aitken; Aitken beat Mieses; Mieses beat Paulsen; Paulsen beat Morphy.)
Louis Morin Louis Morin 8/23/2020 02:40
My Fischer number is 3 (although my current FIDE rating is 1806).

Louis Morin beats IM Lawrence Day (Quebec Open 1984)
Lawrence Day beats Pal Benko (NY Continental Open 1980)
Pal Benko beats Bobby Fischer several times

So what do I win?
Flávio Patricio Doro Flávio Patricio Doro 8/19/2020 12:56
A. Lein and L. Shamkovich are number two, as both beat M. Tal. So, quite a few US players should be number three.
Vannatar Vannatar 8/16/2020 07:40
The key to getting a smaller number definitely seems to involve people beating better players when they're older or younger. As someone suggested earlier in the comments, if the requirement was beating someone in their prime (or at least within, say, 100 rating points of their peak rating, then the "Fischer Number" would naturally be larger.

Additionally, the cross-generational nature of the exercise when you use Bobby Fischer as the 0-number (for many of us at least) means the chain HAS TO involve players defeating significantly older opponents or else result in a much higher number.
Queenslander Queenslander 8/16/2020 11:52
My Fischer number is 4:

... GM Mikhail Tal (beat Fischer 4-0 in Bled/Zagreb/Belgrade Candidates 1959)
... GM Aivars Giplis (beat Tal in Latvian Ch'p 1958)
... IM Aleksander Wohl (beat Gipslis at Biel 1996 - in a wonderful game by the way)
... Queenslander (then 2100 or so beat Wohl in New Zealand Ch'p 2003).
Vannatar Vannatar 8/15/2020 11:20
This was a fun exercise. Turns out as a mere 1995 rated club player from Ohio my Fischer Number is four. I'm even closer (just three!) to former champions Tal and Smyslov - and just four and five away from Karpov and Kasparov respectively.

I beat NM Murthy Pappu (Akron Chess Club, March 26, 2016)
NM Murthy Pappu beat GM Anatoly Lein (King’s Island Open, 2006)
GM Anatoly Lein beat GM Mikhail Tal (USSR Championship, Vilnius, 1955)
GM Mikhail Tal beat GM Bobby Fischer (4-0 in Candidates' Tournament, 1959)

Also interesting:
GM Anatoly Lein beat GM Vasily Smyslov (USSR Championship 1966/1967, Tbilisi, Jan. 22)
GM Vasily Smyslov beat GM Anatoly Karpov (USSR Championship 1971, Leningrad, Sept. 27)
GM Anatoly Karpov beat GM Gary Kasparov (Karpov-Kasparov World Championship Match 1984/85, Moscow, Rd 9, Oct. 5)
Michael Jones Michael Jones 8/15/2020 10:10
Lajosarpad's stricter criterion of "prime Fischer" definitely rules out my original 4 - since although Conor Murphy was already 157 ECF (not sure what that equates to in Elo) in 2010, that's still some way short of his peak (to date) of 232. However, at the same Ealing rapidplay tournament I also beat Tumen Buyandalai - who was 205 ECF at the time, only marginally short of his peak of 208. He has beaten Michael Basman (London Rapidplay 2014), Basman beat Pal Benko (Hastings 1975) and Benko beat Fischer several times. That should qualify as a "prime" 4 - Basman was 68 when he lost to Buyandalai, but still (probably - relies on ECF to Elo conversion) within 10% of his peak rating.
Lars Rasmussen Lars Rasmussen 8/15/2020 03:46
My Fischer number is 4.
I beat Mogens Moe, Xtracon Chess open 2018.
Moe beat Eigil Pedersen, Danish Championship Vejle 1967.
Pedersen beat Bent Larsen in a match in 1952.
Larsen beat Fischer, Palma de Mallorca 1970.
gimiller gimiller 8/15/2020 02:35
My winchain is 3; I beat PH Clarke>beat Bent Larsen>beat RJF
herr_doktor herr_doktor 8/15/2020 01:29
In the original article, Professor Christian Hesse asserts that Bobby Fischer himself has the Fischer number 0.
Let's examine this further. Here is an extended quote from 'Endgame' by Frank Brady where a very young Bobby plays against himself: 'Setting up the men on his tiny board, he'd play game after game alone, first assuming the white side and then spinning the board around, with some pieces often tumbling onto the floor......"Eventually I would checkmate the other guy", he chuckled when he described the experience years later.'
So it looks like Fischer himself really has a Fischer number of 1.
Grey Hiker Grey Hiker 8/15/2020 08:02
My Fischer number is 2.

Arthur Bisguier defeated Ficher (July 10 1956). [Bisguier famously defeated Fischer in their first game, drew the second, and lost all 12 of their other games].

I defeated Bisguier in August 1969 (last round of a large open tournament in New Hampshire, August 18 I think, the same weekend as the famous Woodstock music festival).

FM John Timm
dadian dadian 8/15/2020 03:45
herr_doktor: Take it up with Frederic Friedel. He set the criteria.
GreenKlaser GreenKlaser 8/15/2020 02:50
My number is 2. The required wins were against Arthur Bisguier, who defeated Bobby once. Aside from handshake numbers based on chess games, my number is one based on personal contact such as sitting in the front seat of a car and looking at the same foreign chess books at Buschke's.
Lonnie Kwartler
Peter B Peter B 8/15/2020 02:21
@Tomas - I think Carlsen is a 3. Korchnoi is a 1, and many older modern players like Anand, Kramnik, Gelfand etc have beaten Korchnoi so have a Fischer number of 2, giving Carlsen a 3.

Carlsen did play Korchnoi once when he was young, but lost. I think that was his only chance to get a FN of 2.
Flávio Patricio Doro Flávio Patricio Doro 8/15/2020 02:15
My number is five:
I beat Joaquim de Deus Filho in the u18 Sao Paulo state ch., 1976
Joaquim de Deus Filho beat IM Alexandru Segal in the 2nd Copa Itaú, Sao Paulo 1998.
IM Alexandru Segal beat GM A. Lein at Sao Paulo, 1979.
GM Lein beat GM Pal Benko twice (USA ch 1978, Lone Pine 1978)
GM Benko beat Fischer at Portoroz 1958, Buenos Aires 1960 and Curaçao 1962.
connert3 connert3 8/15/2020 01:51
Like many old US players, my Fischer number is 2. I beat an older Herb Avram, who beat a young Fischer. I wonder how many other 2's I might have - I defeated a quite a few players in the 1970s who were good in the 1950s.
herr_doktor herr_doktor 8/14/2020 10:31
Dadian - beating someone who beat a 12-year old kid with the Latvian Gambit somehow doesn't cut it. Think lajosarpad would agree.
dadian dadian 8/14/2020 09:43
I have beaten Viktors Pupols, so my Fischer number is 2.
DavidFriedman DavidFriedman 8/14/2020 03:40
My Fischer number is four: I have beaten FM Carl Boor, who has beaten GM Alex Shabalov, who has beaten GM Arthur Bigsuier, who beat Fischer.

--David Friedman
HubertKnott HubertKnott 8/14/2020 01:44
14 August 2020

On 29 December 1999, I defeated the late Candidate Master Michael Layevskiy in the 35th Greater New York Scholastics Junior-Varsity section. Layevskiy defeated IM Jay Bonin in two tournament encounters (all of the above information comes from publicly available USCF tournament databases).

Bonin defeated Robert Byrne in the 1986 New York Open (notation and game circumstances recorded on pages 24 (466) and 25 (467) of Chess Life, July 1986 issue). Robert Byrne defeated Fischer in the 1965 US Championship (notation in Chess Life, January 1966, vol. 21, no. 1, page 7).

I suppose that this string gives me a Fischer Win-chain number of 4: me (4)-Layevskiy (3)-Bonin (2)-Byrne (1)-Fischer (0). But sadly, I am forced to go against the spirit of the suggestions offered by lajosarpad, since Layevskiy was only a 1411 USCF at the time of his game with me. Lajosarpad would seem to want everyone with a master or "prime playing" status at the time of the game.

In addition, given the rules of this exercise, I cannot count my videotaped skittles draw game against the late Eliot Hearst, who defeated Fischer. Such a string would have given me a Fischer number (but not a win-chain) of 2.

NM Hayoung Wong

PS: I need not mention the usefulness of USCF rating history databases/profiles as valuable sources of information in this exercise.
lajosarpad lajosarpad 8/14/2020 01:03
I believe that beating Fischer when he was a little boy or an elder is not the same as beating Fischer when he was world champion. We could introduce some number, similar to Fischer number, where we only include games where the loser (closer to the target person) was in his prime. Maybe we could compute his/her maximum ELO - 10% and either

his/her ELO at the time of the game >= his/her max ELO - 10%


he/she had higher ELO than his/her max ELO - 10% both before and after the game.

We could call this Fischer prime number (not a prime per se, but any person in the chain was in his/her prime)
Davidx1 Davidx1 8/14/2020 12:58
[Event "St Stefan/Belgrade m"]
[Date "1992.]
[White "Fischer, Robert James"]
[Black "Spassky, Boris V"]
1. e4 e5 2. Nf3 Nc6 3. Bb5 a6 4. Ba4 Nf6 5. O-O Be7 6. Re1 b5 7. Bb3 d6 8. c3
O-O 9. h3 Nb8 10. d4 Nbd7 11. Nbd2 Bb7 12. Bc2 Re8 13. Nf1 Bf8 14. Ng3 g6 15.
Bg5 h6 16. Bd2 Bg7 17. a4 c5 18. d5 c4 19. b4 Nh7 20. Be3 h5 21. Qd2 Rf8 22.
Ra3 Ndf6 23. Rea1 Qd7 24. R1a2 Rfc8 25. Qc1 Bf8 26. Qa1 Qe8 27. Nf1 Be7 28.
N1d2 Kg7 29. Nb1 Nxe4 30. Bxe4 f5 31. Bc2 Bxd5 32. axb5 axb5 33. Ra7 Kf6 34.
Nbd2 Rxa7 35. Rxa7 Ra8 36. g4 hxg4 37. hxg4 Rxa7 38. Qxa7 f4 39. Bxf4 exf4 40.
Nh4 Bf7 41. Qd4+ Ke6 42. Nf5 Bf8 43. Qxf4 Kd7 44. Nd4 Qe1+ 45. Kg2 Bd5+ 46. Be4
Bxe4+ 47. Nxe4 Be7 48. Nxb5 Nf8 49. Nbxd6 Ne6 50. Qe5 1-0
Learn to ride,people ...
Frits Fritschy Frits Fritschy 8/14/2020 12:44
Two! I beat Donner in a simul in 1971.
OR_Log OR_Log 8/14/2020 12:20
My Fischer-number is 4: I beat Sauli Tiitta (3) who beat Ilkka Kanko (2) who beat Jan-Hein Donner (1) who beat Fischer 1962.
mtm57 mtm57 8/14/2020 12:20
Fischer is always the most interesting champ
KnightOnTheRim KnightOnTheRim 8/14/2020 09:55
3 , via a win vs Lautier who has beaten Kortschnoi
perstreperous perstreperous 8/14/2020 09:44
4, although the start of the chain was a simultaneous.

I beat Roddy McKay (in a simultaneous) who beat Vlastimil Hort (1982 Olympiad, a famous game) who beat Boris Spassky (twice) who beat Bobby Fischer (eleven times).
Tomas Yttling Tomas Yttling 8/14/2020 08:42
Anybody know Carlsens Fischer number? I am 3 from Carlsen in OTB w/ standard time control (Roland Eriksson - IM Emil Hermansson - Carlsen), and I figure that's got to the shortest path possible for me to achieve.
Peter B Peter B 8/14/2020 07:20
I am lucky enough to have wins against a couple of former Australian champions, Alex Wohl and Doug Hamilton. They each have a Fischer number of 3, giving me a Fischer number of 4.
* Wohl beat Miles (1991 Australian Open), who beat Korchnoi a few times, who beat Fischer.
* Hamilton beat Gacharna (1970 Olympiad), who beat Korchnoi (1962 Interzonal), who beat Fischer.
Hamilton was also a piece up against Korchnoi at the 1970 Olympiad, but unfortunately couldn't force a win and give himself a "2" (and me a "3").
Boisgilbert Boisgilbert 8/14/2020 01:28
I beat Josh Manion who beat Andrew Soltis who beat Arthur Bisguier who beat Bobby Fischer, so 4. Or if you count simuls and blitz, I beat Karl Zangerle (blitz) who beat Bent Larsen (simul) who beat Bobby (3).
moontan moontan 8/14/2020 01:00
My number would be 2: I beat Milton Otteson in the 1963 Minnesota St Ch (at the Univ of Minn Twin Cities campus in February), who beat Fischer in the 1957 Western Op (in Milwaukee, WI).
herr_doktor herr_doktor 8/13/2020 11:08
My own Fischer number is 4.
I once beat FM Neil Berry of Scotland, who beat IM Craig Pritchett (also of Scotland) in 2015, who beat
GM Florin Gheorghiu in 1980, who beat Bobby in Havana, 1966.
Michael Jones Michael Jones 8/13/2020 10:51
Quite surprised to find that my Fischer number is only 4:

I beat Conor Murphy, Ealing Rapidplay 2010
Conor Murphy beat Keith Arkell, Steve Boniface Memorial Congress 2019
Keith Arkell beat Robert Byrne, Watson, Farley & Williams Tournament 1991
Robert Byrne beat Bobby Fischer, US Championships 1965

Same distance to Kramnik: for the last two links, substitute Arkell beat Michael Adams, Hastings 1995, and Adams beat Vladimir Kramnik several times.
chippan chippan 8/13/2020 10:00
I drew with Bob Wade who drew with Fischer. 1 point for a win, half for a draw so my tongue-in-cheek Fischer score is 0.5 + 0.5 = 1.