What years were 2 cent postcards used in the US?
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The US Large Cent was disconinued in 1857. The Flying Eagle type small cent was minted from 1856 to 1858. The Indian Cent was discontinued in 1909. The Lincon cent is still in production today although it has been redesigned every 50 years since 1909. The US 2 cent coin was minted from 1864 to 1873.
The first coins called pennies were made in Britain over 1000 years ago so it's not possible to know exactly when and where the first one was struck. US 1¢ coins are technically cents; the slang "penny" is a holdover from colonial times when British coins were used. The first circulating US cent was struck in 1793, although experimental and pattern cents (e.g. the famous Fugio cent and silver-center cent) were minted several years earlier.
You may want to review your coin again - the US Treasury did not mint its frst cent until 1793 (Flowing Hair Large cent, Chain Reverse), followed by the Liberty Cap Large cent (1793-1796). You may have a Treasury medallion. These are struck on cent blanks and have been included in US coin sets for many years. There's one for each Mint. They have no special value because they're the same each year.