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"Why not to make the game get harder each minute? Or you change weapon type each minute"

Those are good ideas, I guess you could also do score multipliers every minute and so on as well. Or challenges for under a minute that give you a boost or something.

Then there is also bonus/power-up lasts for one minute, or even a sports game with 1 minute timer for rounds or qualifying etc.

Ooh, single minute sports sounds super neat.  I might try that!