I'd like to play the original Tetris I played near the end of the 80's, the Arcade game:
Does anyone know exactly what this version of Tetris is called?
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You can play it with MAME (Multiple Arcade Machine Emulator). This emulator have open source code, you can download build or sources here.
(as questions about direct links to ROM are considered as an offtopic, I can't provide such link)
More info here.