He played ten times in Soviet Championships. In 1929, he took 3rd in Odessa (6th URS-ch; Boris Verlinsky won). In 1931, he took 7th in Moscow (7th URS-ch; Mikhail Botvinnik won). In 1933, he took 9th in Leningrad (8th URS-ch; Botvinnik won). In 1934/35, he tied for 9-12th in Leningrad (9th URS-ch; Grigory Levenfish and Ilya Rabinovich won). In 1937, he took 13th in Tbilisi (10th URS-ch; Levenfish won). In 1939, he tied for 13-14th in Leningrad (11th URS-ch; Botvinnik won). In 1945, he took 17th in Moscow (14th URS-ch; Botvinnik won). In 1947, he tied for 13-15th in Leningrad (15th URS-ch; Paul Keres won). In 1952, he took 18th in Moscow (20th URS-ch; Botvinnik and Mark Taimanov won). In 1955, he took 17th in Moscow (22nd URS-ch; Vasily Smyslov and Efim Geller won).
In the 1931 Moscow City Championship, he took 2nd, behind Nikolai Riumin. In the 1932 Moscow City Championship, he took 3rd, behind Nikolai Riumin and Duz-Hotimirsky.  In 1933/34, he took 5th in Moscow–ch (Riumin won). In 1934, he took 5th in Leningrad (Botvinnik won). He tied for 6-7th at Moscow 1935 (2nd it; Botvinnik and Salo Flohr won). In 1936, he tied for 7-10th in Moscow (3rd it; José Raúl Capablanca won). In 1936, he tied for 1st-2nd with Vladimir Alatortsev in Moscow–ch. In 1937, he took 4th in Moscow–ch (Alatortsev and Sergey Belavenets won). In 1937, he took 2nd, behind Reuben Fine, in Moscow.
He was awarded the IM title in 1950. His opening theory contributions were mainly in the Sicilian Defence variation which bears his name: 1.e4 c5 2.Nf3 e6 3.d4 cxd4 4.Nxd4 a6.  This flexible line has been adopted frequently by top players since 1950, including Anatoly Karpov, World Champion 1975-85, and English Grandmaster Tony Miles. The line is sometimes called the Paulsen Variation in the West, for German Master Wilfried Paulsen, who played it earlier in the 19th century. Kan scored several victories over Botvinnik early in his career; Botvinnik became World Champion 1948–1957, 1958–1960, and 1961–1963.
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