Donald Knuth, basketball and computers in sport


Donald Knuth received a scholarship in 1956 to attend the Case Institute of Technology in Cleveland, Ohio. During his time at the Institute, Donald was manager of the basketball team. His interests in observation, notation and computer programming were recorded in an IBM documentary.


At Case, Donald combined his interest in computers with his involvement in basketball. He had managed his High School basketball team and took on this role at Case during his undergraduate years (1956-1960).

The Computer History Museum‘s biography of Donald includes this paragraph:

Knuth’s lifelong love affair with computers began as an undergraduate when he discovered the IBM 650 computer system at Case. He quickly mastered the inner workings of the machine and developed a novel program to automate coaching of the school’s basketball team, earning him an appearance on the CBS Evening News with Walter Cronkite.

This novel program is reported in a variety of contexts … all of them I missed in my research into computers in sport.

Donald’s Basketball System

The Internet Archive has a copy of Donald’s chapter 23 in Selected Papers in Fun and Games. In it he wrote:

In high school I’d come up with a general rule of thumb that said,“ Possession of the ball is worth roughly one point, except near the end of a period. ” In other words, if you enter the stadium at a time when your team is leading by a score of 50~49, the effective score is really 51-49 if your team has the ball, but the game is basically tied if the other guys have possession. A corollary of this rule is that field goals don’t really change the effective score! One team gains 2 points, but loses possession, while the opponents gain possession. The score really changes when there’s a turnover, or when a free throw is made, or at the very end.

Of course I knew that this rule of thumb was only a rough approximation; maybe possession was worth only .8 of a point, say. Even so, the person who steals the ball should be rewarded more than the person who makes baskets, contrary to the normal way that players get credit for their contributions.

At Case, “I finally had an opportunity to test these hunches in a quantitative way, and the computer program I wrote was based on those informal notions about possession. To everyone ’s surprise, including my own, the system turned out to be quite successful.”

Chapter 23 provides a detailed account of Donald’s observation system. The records were kept in an “electronic computer”.

The statistics at each basketball game can be taken by two men: a recorder and a spotter. After the game it takes approximately 30 minutes to prepare the necessary totals from the game sheets and about three minutes to punch the IBM cards. One IBM card is made for each Case player who participated in the game, plus a card each for Case and the opponents. Then the machine takes 1.5 minutes to process the game: 30 seconds to take in the “program” of instructions for calculation, 30 seconds to take in the statistical data from all the previous games, and 30 seconds to take in the statistics from this game and to punch the answers. The computer punches four cards for each player, two indicating his performance in this particular game and two containing his cumulative record to date. The cards can easily be printed up for reference and can be filed neatly. Any desired set of statistics can quickly be found from them by passing them through a sorter.

Donald’s real-time notation recorded:

    • Field goals attempted and made (divided into short, medium, and long range).
    • Total free throws attempted and made.
    • Last free throws of a set, made and missed.
    • Total fouls and offensive fouls.
    • Rebounds, defensive and offensive.
    • Violations of rules causing loss of ball.
    • Assists.
    • Loss of ball by fumble, bad pass, or jump ball.
    • Gain of ball by interception or jump ball.
    • Defensive mistakes — allowing opponent to score a field goal.
    • Minutes played.

Donald recorded these data in real time on pre-printed observation sheets. He collected all eleven elements of these data for each Case player and the first six elements for the opposing team.

His program gave each Case player a personal score. Donald noted:

The computer calculates the “true point contribution” of a player by using a rather complicated formula. Using the abbreviations FGA (field goals attempted), FGM (field goals made), FTA (free throws attempted), FTM (free throws made), LFTI (last free throws made), LFTO (last free throws missed), TF (total fouls committed), OF (offensive fouls), OR (offensive rebounds), DR (defensive rebounds), VIOL (violations), AST (assists), FUM (fumbles), BP (bad passes), JL (jump balls lost), JG (jump balls gained), INT (interceptions), DM (defensive mistakes), the player’s “point contribution” rating is:

PC = 2FGM + FTM + 2(AST – DM)

– a(VI0L + FUM + JL + BP + AST + FGM + LFTI)

+ /3(INT + JG + OR + DR + DM – OF)

– 7 (TF) – S (FGA – FGM + LFTO),

where a, f3, 7 , and 5 are weighting coefficients determined by team totals. In the formulas for these coefficients, small letters indicate opponents’ totals and capital letters denote Case totals:

a = 2f gm/ (f ga + viol + of – or + INT + JG + TF – OF);

/3 = 2FGM/(FGA + VIOL + OF – OR + FUM + JL + BP + tf – of);

7 = (ftm – /3(lfti + If to x DR/ (or + DR))) /TF:

6 = a x dr/(0R + dr).

The Archive record includes a copy of Donald’s mimeographed form.

Donald said of his program as an afterthought:

Alas, I have been unable to find any copies of the original program, nor do I recall who carried on with it after I left for graduate school.

My formula for PC should not be taken too seriously. I kept fiddling with it, and never really believed that it was rigorously correct. This work was done long before I had ever heard of Markov processes.

I communicated details of this work to some people at Marquette University in the early 1960s. But it has almost surely had no influence on subsequent applications of computers to sports, except perhaps to stimulate others to do better.

By 1995, professional basketball teams were using computers routinely … (My emphasis)

Analysing Performance

Donald started his basketball analysis at Case fourteen years after Lloyd Messersmith’s dissertation was submitted for examination. In the intervening years there had been significant breakthroughs in computational methods.

A decade after Donald’s experiences at Case, another computer scientist Anatolij Zelentsov was exploring the use of computers in the analysis of association football. This was another iteration in the computerised notation and analysis of performance that provided a ‘functional readiness’ measure for each player coached by Valerij Lobanovs’kyj at Dynamo Kiev.

We are fortunate that we have Donald’s account of his work. You might find Chapter 11 of Donald’s interview on the Web of Stories of particular interest. In it he describes in detail his work at Case.

I found it fascinating to listen to the reflections of his early days in computing after a lifetime of engagement in computer science and the author of The Art of Computer Programming.

I am profoundly sorry it has taken me so long to write about his work. I hope it adds to our knowledge of our origins.


The New York Times wrote about Donald’s work in computer science in December 2018. The title of the article was The Yoda of Silicon Valley (link).

Valerij, Anatolij and Dynamo Kyiv’s Golden Years



I have been researching Valerij Lobanovs’kyj’s coaching career at Dynamo Kiev. I am particularly interested in his partnership with the statistician and analyst Anatolij Zelentsov.

I believe their partnership is a defining moment (that extended over three decades) for those of us involved in the analysis of performance and the quest for actionable insights.

There are two parts to their partnership at Dynamo Kyiv: 1974-1990; and 1997-2002.

Over these two periods, Dynamo Kyiv’s goals scored and conceded were:

The gap in the data between seasons 18 and 19 signal Valerij’s absence from the club. When he left at the end of the 1990 season, Dynamo Kyiv were in the USSR Vysshaya Liga. On his return in 1997, the team were in the Ukraine Premyer Liga.

1974-1990 Vysshaya Liga (USSR)

In the eighteen seasons Valerlij and Anatolij were together at Dynamo Kiev, the club won: eight Vysshaya Liga titles ( 1974, 1975, 1977, 1980, 1981, 1985, 1986, 1990); six Soviet Cup finals (1974, 1978, 1982, 1985, 1987, 1990); two UEFA Cupwinners’ Cup competitions (1975, 1986); and the UEFA Super Cup (1975).

In 1976, the Vysshaya Liga was divided into two seasons (Spring and Autumn). The club finished seventh in the Spring and second in the Autumn. In that year Valrij coached the Russian Olympic football team that won a bronze medal at the Montreal Olympics. Valerij coached the Russian national team in the 1983 and 1984 seasons. Dynamo Kiev had their worst two years in the Liga since Valerij had arrived (7th in 1983, 10th in 1984). When he returned in 1985, Dynamo Kyiv won the league title and did so again the following year.

In 1974-1975, Dynamo Kyiv defeated CSK Sofia, Eintrach Frankfurt, Bursaspor, PSV Eindhoven, and Ferencvaros (3v0 in Basel) to with the UEFA European Cupwinners’ competition.

In 1985-1986, their opponents were: Utrecht, Craiova (100,000 spectators saw the second leg at Kyiv), Rapid Wien, Dukla Prague, and Atletico Madrid.  Dynamo Kyiv won the final 3v0 in Lyon.

1997-2002 Premyer Liga (Ukraine)

Valerij returned to Dynamo Kyiv at the start of the 1997-1998 season. At this time the team were in the Ukrainian Premyer Liga. In Valerij and Anatolij’s second partnership at the club, Dynamo Kyiv won four consecutive league titles (1997-1998, 1998-1999, 1999-2000, 2000-2001); and three Ukraine Cup Finals (1998, 1999, 2000).

Valerij’s last game was away against Metalurg Zaporizhya on 7 May 2002. He suffered a stroke shortly after the game and died a week later on 13 May. Dynamo had won the game 3v1 and were leading the League by four points. In that season they had won 17 of the 21 games they had played, were undefeated and had scored 52 goals and conceded 7. Their only defeat that year came in round 25 when Shakhtar Donetsk beat them 2v0 to win the title by one point.

Anatolij died four years later. He was still working with the club at the time of his death.

Greater than the sum of their parts

I have been fascinated by a friendship that started in 1968 and only ended with Valerij’s death in 2002. This friendship combined a very special coach and an analyst who was around at the time it became possible to use computers in sport. Together they took part in a golden age of Dynamo Kyiv football and did so in two distinct phases of their careers.

I thought this quote, attributed to Anatolij, embodies what this friendship meant:

Ideas are good, but most important is to realise them in practice. Valerij is the unsurpassed master in the realisation of ideas. What’s more, he does it in his own way. (GOTP, 2015)

Photo Credits

Dynamo Kyiv 1975 (Game of the People, 2 November 2015)

The group of four (Pasquale)

Valerij Lobanovs’kyj (Alchetron)

Anatolij Zeletsov obituary (Dynamo Kiev)


This is my fourth post about Valeij and his coaching career. The four posts are my belated attempt to research a remarkable part of football history that has relevance for coaches and analysts. The limitations of language have prevented me from pursuing granular detail in this research.

The posts combine three of my passions: coaching; analysis and analytics; life history.

Valerij, Anatoli, Oleh and Mykhailo

Valerij Lobanovs’kyj

Anatolij Zelentsov

Anatolij Zelentsov


It is hard to find pictures of Anatolij Zelentsov. He was the statistician and analyst who accompanied Valerij Lobanovs’kyj on his coaching journey with Dnipro Dnipropetrovsk and subsequently at Dynamo Kyiv. This portrait picture is from  the Komkon website.

As part of my ongoing exploration of the golden age of Dynamo Kyiv under Valerij’s leadership, I wanted to share what I have learned about Anatolij (I am going to spell his first name to resonate with Valrij, I think his name in Ukranian is Анатолій Зеленцов).

Simon Kuper (Football Against the Enemy) and Jonathan Wilson (Behind the Curtain) have shared their experiences of meeting Anatolij in person. Their accounts started my research about Anatolij. Jonathan (2012) described Anatolij as ““a young academic brimming with enthusiasm for the statistical methods he believed could be employed to improve standards of coaching”.

Dnipropetrovsk Beginnings

In an interview with Komkon, Anatolij remembers being introduced to Valerij “sometime” in 1968. At that time he was the Dean of the Dnipropetrovsk Institute of Physical Science, Valerij was about to become head coach of the Dnipro Dnipropetrovsk football team.

Dnepropetrovsk (Dnipro since 2016) is approximately 400 kilometres from Kiev. At the time Valerij and Anatolij met, Kiev was the centre of the Soviet computer industry. Jonathan Wilson (2011) points out:

The first cybernetic institute in the USSR was opened there in 1957 and quickly became acknowledged as a world leader in automated control systems, artificial intelligence and mathematical modelling. It was there in 1963 that an early prototype of the modern PC was developed.

Anatolij gave Valerij access to this cybernetics culture and, as Gabrielle Marcotti (2013) points out, they came of age together:

when the Soviet Union was developing its first computers. Yes, they were the size of minibuses and, in terms of computing power, roughly equivalent to a convenience store microwave.

They were at the forefront of the use of computers in sport “at a time when the most advanced technology used by coaches around the world consisted of a whistle, a legal pad and a ballpoint pen” (Gabrielle Marcotti, 2013).

The Development of Training Models

Anatolij and Valerij shared their thoughts about performance in their book The Methodological Basis of the Development of Training Models. (David Squires (2016) suggests “they were too busy decrypting football to think of a punchier title”.)

I was not able to find an online source for the book. A number of authors cite short passages from it. In his interview with Komcon, Anatolij reports “This book was censored for a long time. They demanded to remove “anti-sovietism”. He added:

Our first thoughts about modelling of the game appeared exactly in a theatre. We, with Lobanovsky, used to come to watch rehearsals in a theatre, to see how the future show is “modelled”.

One of the most quoted parts (see, for example, Jonathan Wilson (2011); Vadim Furmanov (2012); and Jed Davies (2013)) of the book is:

When we are talking about tactical evolution, the first thing we have in mind is to strive for new courses of action that will not allow the opponent to adapt to our style of play. If an opponent has adjusted himself to our style of play and found a counterplay, then we need to find new a new strategy. That is the dialectic of the game. You have to go forward in such a way and with such a range of attacking options that it will force the opponent to make a mistake. In other words, it’s necessary to force the opponent into the condition you want them to be in. One of the most important means of doing that is to vary the size of the playing area.

Within this paragraph, we have socialism meeting dynamical systems or as Vadim Furmanov (2012) suggests “Scientific Communism meets Total Football”. This approach led Anatolij to evaluate the functional readiness of players.

In an interview in 2004, Anatolij indicated where his functional evaluations had taken him:

In the 50-60s, many coaches believed that the more a team trains, the better it plays. However, you can spend three hours on the field, without getting the proper load, but you can work fruitfully for 45-50 minutes. The basis was to increase the intensity of training with a decrease in their duration. In this case, it was necessary to take into account the specifics of the recovery period, theoretically simulate and practically justify the model of the training session with the known reactions of the player’s organism in advance. To work scientists were connected with developments in related fields – Academician Glushkov, Professors Zimkin, Chagovets, Yakovlev. Together, we achieved results that had no analogues in the world at that time.


Barney Ronay observed recently:

Lobanovskiy is usually cast as the father of things. Father of analytics. Father of a data-driven total football. … every movement tracked, rated and tessellated.

Tessella‘ is a Latin word for a tile or small square. It is also used to describe patterned mosaic floors. Tessellation is at the heart of the system Anatolij and Valerij developed.

Many of the insights into this tessellation come from Italian discussions of Anatolij’s work. For example, a post by Laboratorio Pincharrata (2012) quotes Anatolij:

In my laboratory the functional disposition of the players is evaluated and how they can reach their best potential. We develop the players in a natural way following the scientific recommendations We recommend how to plan the training, how to evaluate them, how to understand the actions of the players in the field, all from a scientific point of view.

Anatolij’s computer program for the evaluation of ‘functional readiness’ to analyse game play and individual player performance divided the playing field into nine squares. Within this tessellation, Anatolij measured the frequency with which each player entered defending and attacking areas and the work he did on and off the ball. Each player was awarded an overall score to three decimals based on physical, cognitive and affective measures.

The only example of an output of this system I have been able to find appeared in the Laboratorio Pincharrata (2012) post:

Anatolij said of this approach:

In our lab we are trying to evaluate the potential of every player. Moreover, we don’t just give coach an advice, we justify it with numbers. We recommend how to compose the program of trainings, how to evaluate it, how to understand actions of players on the field… all from the scientific point of view, no emotions. (Komokon, nd)

He added that the laboratory had a decade of records:

Tracing all players — those who leave the team and those who remain with us. It doesn’t matter for us. We developed different methods of training, different models of games. We have have fast video-database with almost all tournaments in the world. Starting from them we are trying to predict the development direction of Modern Football. (Komokon, nd)


The story of Dynamo Kiev’s golden years raises some fascinating insights for present day analysts as they work behind the scenes with coaches and players.

Valerij had a vision for how the game of football might be played. His technical and tactical coaching insights pervade present day football, fifteen years after his death in 2002. One of his observations was “A system does not guarantee success, but it gives a much better chance of success than making it up as you go along’.

With Anatolij as his partner, he was able to develop a system that transformed preparation and game play into chess rather than an unthinking acceptance of chance. The system involved 22 on-field elements transformed by the understanding of game play as a dynamical (dialectic) flow. Anatolij’s view of this was “continuous motion of players, collective speed actions of a team, high individual skills based on the foundation of collective actions” (Komokon, nd).

Throughout my research for this post, I have been struck by the impact of a long-term friendship on transforming performance. I think Anatolij and Valerij’s relationship was a perfect meeting of minds. From this grew an exemplary synergy.

I am mindful that their meeting has been referred to as ‘accidental’ which is to me the other part of very special relationships … the magic of serendipity.


Anatolij died in September 2006 at the age of 72. I found an obituary for him using his name in Ukrainian (Анатолій Зеленцов). The Dynamo Kiev website noted:

Almost forty years of his life, Anatolij Zelentsov devoted to “scientific football”, creating an original system of accurate calculation of the training process, mathematical modeling of loads.


Until the last day of his life, Anatolij Zelentsov did not stop in development, giving the updated training staff of Dynamo and the Ukrainian team the latest scientific developments in the field of increasing the intensity of training sessions, models of playing football and much more.

The notice listed three publications (in addition to The Methodological Basis noted above):

The experience of programming the preparation of the players of the Dynamo Kyiv team, the programming of the training of football teams of masters (with Valerij Lobanovsky)
Tactics and strategy in football (with Valerij Lobanovsky, V. Tkachuk, A. Kondratiev)
Lessons of football (with Valerij Lobanovsky, V.Kowerver, V.Tkachuk)

Photo Credits

Anatolij Zeletsov (Komkon)

The group of four (Pasquale)

Dynamo Kiev 1985-87 (FourFourTwo)

Anatolij Zeletsov obituary (Dynamo Kiev)