Etymology
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baseball (n.)

in the modern sense of a game of ball for teams of nine, 1845, American English, from base (n.) + ball (n.1). Earlier references, such as in Jane Austen's "Northanger Abbey," refer to the game of rounders, of which baseball is a more elaborate variety. The modern game was legendarily invented 1839 by Abner Doubleday in Cooperstown, N.Y. Base was used for "start or finish line of a race" from 1690s; and the sense of "safe spot" found in modern children's game of tag can be traced to 15c. (the use in reference to the bags in modern baseball is from 1868). Baseball as "ball with which the game of baseball is played" is by 1885.

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game (adj.1)
"lame," 1787, from north Midlands dialect, of unknown origin, perhaps a variant of gammy (tramps' slang) "bad," or from Old North French gambe "leg" (see gambol (n.)).
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game (v.)
Middle English gamen "to sport, joke, jest," from Old English gamenian "to play, jest, joke;" see game (n.). The Middle English word is little recorded from c. 1400 and modern use for "to play at games" (1520s) probably is a new formation from the noun; and it might have been re-re-coined late 20c. in reference to computer games. Related: Gamed; gaming.
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game (n.)

c. 1200, from Old English gamen "joy, fun; game, amusement," common Germanic (cognates: Old Frisian game "joy, glee," Old Norse gaman "game, sport; pleasure, amusement," Old Saxon gaman, Old High German gaman "sport, merriment," Danish gamen, Swedish gamman "merriment"), said to be identical with Gothic gaman "participation, communion," from Proto-Germanic *ga- collective prefix + *mann "person," giving a sense of "people together."

The -en was lost perhaps through being mistaken for a suffix. Meaning "contest for success or superiority played according to rules" is first attested c. 1200 (of athletic contests, chess, backgammon). Especially "the sport of hunting, fishing, hawking, or fowling" (c. 1300), thus "wild animals caught for sport" (c. 1300), which is the game in fair game (see under fair (adj.)), also gamey. Meaning "number of points required to win a game" is from 1830. Game plan is 1941, from U.S. football; game show first attested 1961.

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game (adj.2)
"ready for action, unafraid, and up to the task;" probably literally "spirited as a game-cock," 1725, from game-cock "bird bred for fighting" (1670s), from game (n.) in the "sport, amusement" sense. Middle English adjectives gamesome, gamelich meant "joyful, playful, sportive."
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game-cock 
cock bred for fighting or from fighting stock, 1670s, from game (n.) in the sporting and amusement sense + cock (n.1). Figurative use by 1727.
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board-game (n.)
also boardgame, 1867, from board (n.1) + game (n.). Compare German Brettspiel.
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no-hitter (n.)
baseball term for a baseball game in which one side fails to make a hit, 1939, from no + hit (n.).
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softball (n.)
baseball of larger than usual size, used in a scaled-down version of the game, 1914, from soft + ball (n.1). The game itself so called from 1916, also known as playground baseball. The word earlier was a term in sugar candy making (1894). Softball question, one that is easy to answer, is attested from 1976.
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tag (v.2)
"a touch in the game of tag," 1878; in baseball, 1904, from tag (n.2); the adjective in the pro-wrestling sense is recorded from 1955. Related: Tagged; tagging.
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