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Gary Kasparov's reign as world chess champion is assured for another three years. As for the Soviet domination of the game, there is no end in sight.

Even as Kasparov and Anatoly Karpov battled, eliminations were under way to determine the next challenger in 1993. As usual, the Soviets have achieved a disproportionate share of success so far.Preliminary tournaments narrowed the field to 15: seven Soviets, one defector who lives in Switzerland, two others from Eastern Europe (Yugoslavia and Hungary), and just five from the rest of the world (two Britons plus one each from the Netherlands, Germany and India).

With title matches held in three-year cycles, the United States is already out of contention until 1996 at the earliest - but that's hardly unusual.

The Soviets have monopolized the game so completely since World War II that except for American Bobby Fischer's brief reign from 1972 to 1975, no player born in another country has even reached the position of challenger.

Some reasons for Soviet domination: a long tradition in the game; fewer competing attractions than in the West; a society that glorifies chess accomplishments; and government support for training promising players.

Some experts feel that up-and-coming young Soviets, like Boris Gel-fand and Vassily Ivanchuck, may be able to defeat Karpov, 39, in qualifying rounds over the next three years. Both are under 22 and ranked third and fourth in the world, behind Kas-parov and Karpov.

Neither Kasparov nor Karpov is likely to retain more than one-fourth of his winnings after the Soviet state takes its share - "Too much," both men agree.

After the formal winning ceremonies, Kasparov returned to Moscow. He is scheduled to visit the United States at the end of the month to promote chess.

For those who keep scrapbooks of world championship tournaments, here are some statistics to add to the list.

- Here are the game-by-game results. The first 12 games were played in New York, and the final 12 in Lyon, France. Each game is listed in this order: game, date, white defense, moves, result, score.

Game 1: Oct. 8, Karpov, King's Indian, 30, draw, .5-.5

Game 2: Oct. 10, Kasparov, Ruy Lopez, 44, Kasparov 1.5-.5

Game 3: Oct. 15-16: Karpov, King's Indian, 53, draw, 2-1

Game 4: Oct. 17-18: Kasparov, Ruy Lopez, 41, draw, 2.5-1.5

Game 5: Oct. 22, Karpov, King's Indian, 36, draw, 3-2

Game 6: Oct. 24, Kasparov, Ruy Lopez, 42, draw, 3.5-2.5

Game 7: Oct. 26-27, Karpov, King's Indian, 44, Karpov, 3.5-3.5

Game 8: Oct. 29-30, Kasparov, Ruy Lopez, 84, draw, 4-4

Game 9: Oct. 31, Karpov, Grunfeld, 34, draw, 4.5-4.5

Game 10: Nov. 2, Kasparov, Petrov, 18, draw, 5-5

Game 11: Nov. 5, Karpov, King's Indian, 24, draw, 5.5-5.5

Game 12: Nov. 7, Kasparov, Ruy Lopez, 37, draw, 6-6

Game 13: Nov. 24-25, Karpov, Grunfeld, 41, draw, 6.5-6.5

Game 14: Nov. 26-27, Kasparov, Scotch, 40, draw, 7-7

Game 15: Nov. 28, Karpov, Grunfeld, 33, draw, 7.5-7.5

Game 16: Dec. 1-2-3, Kasparov, Scotch, 102, Kasparov, 8.5-7.5

Game 17: Dec. 5, Karpov, Grunfeld, 40, Karpov, 8.5-8.5

Game 18: Dec. 8-9, Kasparov, Spanish, Kasparov, 9.5-8.5

Game 19: Dec. 10, Karpov, King's Indian, 39, draw, 10-9

Game 20: Dec. 15-16, Kasparov, Ruy Lopez, 41, Kasparov, 11-9

Game 21: Dec. 19-20, Karpov, King's Indian, 86, draw, 11.5-9.5

Game 22: Dec. 26, Kasparov, Ruy Lopez, 43, draw, 12-10

Game 23: Dec. 29, Karpov, King's Indian, 30, Karpov, 12-11

Game 24: Dec. 31, Kasparov, English Opening, 36, draw, 12.5-11.5

- Here is a list of the past world champions and the years they held the title:

1866-94 - Wilhelm Steinitz, Austria

1894-1921 - Emmanuel Lasker, Germany

1921-27 - Jose Capablance, Cuba

1927-35 - Alesander Alekhine, France

1935-37 - Max Euwe, The Netherlands

1937-46 - Alesander Alekhine, France

1948-57 - Mikhail Botvinnik, Soviet Union

1957-58 - Vassily Smyslov, Soviet Union

1958-59 - Mikhail Botvinnik, Soviet Union

1959-61 - Mikhail Tal, Soviet Union

1961-63 - Mikhail Botvinnik, Soviet Union

1963-69 - Tigran Petrossian, Soviet Union

1969-72 - Boris Spassky, Soviet Union

1972-75 - Bobby Fischer, United States

1975-85 - Anatoly Karpov, Soviet Union

1985-present - Gary Kasparov, Soviet Union

- CONGRATULATIONS TO THE SOLVERS! - Ted Pathakis, Gordon Green, Hal Harmon, Edwin O. Smith, Robert W. Lee, William De-Vroom, Alison Hermance, Ashley Ann Graves, Stanley Hunt, Raeburn Kennard, Nathan Kennard, Aaron T. Kennard, David Moody, Kay Lundstrom, Russell Anderson, Jim Reed, Eugene Wagstaff, Jack Crandall, Ardean Watts, Hal Knight and Glannin Cloward.