From what I can see going through the seasons all the way back to 1890, the Buckeyes have had a total of 14* seasons with a losing record. The years with losing records are 1890 (1-3-0), 1893 (4-5-0), 1894 (5-6-0), 1897 (1-7-1), 1898 (3-5-0), 1922 (3-4-0), 1923 (3-4-1), 1924 (2-3-3), 1943 (3-6-0), 1947 (2-6-1), 1959 (3-5-1), 1966 (4-5-0) & 1988 (4-6-1). I have an asterisk because of the 12 wins vacated from the 2010 season, putting the record for 2010 at 0-1-0.
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In 1897, Ohio State lost 7 games. There have been a few occasions when they lost 6.