Saturday, July 27, 2002
M I N D  G A M E S

The rook and pawn endgame

In many cases, mathematics is an escape from reality. The mathematician finds his own monastic niche and happiness in pursuits that are disconnected from external affairs. Some practice it as if using a drug. Chess sometimes plays a similar role.

Stanislaw Ulam

"FOR all their wealth of content, for all the sum of history and social institution invested in them, music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery," says G. Steiner.

Mr Spock of chess, master of logic, tutor of Garry Kasparov, father of modern "Soviet" chess, Mikhail Botvinnik, whose method was tireless pre-match research, once said that chess was the art that complemented the science of logic, just as music complemented acoustics, and painting optics, etc. In 1948, he became the World Champion, a post made vacant by the death of Alekhine in 1946. Botvinnik relinquished the world title on two occasions, only to win it back in revenge matches: the only player ever to have achieved this.


Few know that there have been Amateur World Champions as well. This was the title awarded in early Chess Olympiads to the winner of an individual tournament in which only amateur players competed. Mattison of Latvia was the first to win the title in the unofficial 1924 Paris Chess Olympiad. The only other holder of the title was Max Euwe, who won it at the 1928 Chess Olympiad, after which, the title was abolished. This underrated player produced some of the most remarkable upsets in the Chess World Championships. I am sure that you still remember the endgame puzzle that he gave to Alexander Alekhine, master blaster of chess, some weeks ago.

Let eij be the result of the game between the i-th and the j-th player.

eij={ 1 if i-th player wins; -1 if j-th player wins; 0 if both have a draw or i=j}

Is it possible that for every particular player, the sum of points of the players who were beaten by him is greater than the sum of points of the players who beat him? This means: is it possible that:


eij C j >0 " i


where C j is a score of the j-th player. Multiplying these inequalities by C i and summing up for all i, we find that


eij C i C j >0


which is impossible, since the left hand expression is 0 because eij = -eji for all i, j.

For every particular player, the sum of points of the players who were beaten by him is, thus, neither greater nor less than the sum of points of the players who beat him.

Master Mayur Bhatt of Sujanpur Tihra and Pawan Goyal answer "no", but give no reason for it as well. "No, it is not possible, because it is true only for the players who beat all other players," says Sudhanshu Arya. Honey Deep says that it is possible (checkmate!). Write at The Tribune or [email protected] *DELAYED BUS: Correct solution to the 'Japanese chip' problem was also received from Mohit Bhambri of Patiala.

Aditya Rishi