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How many wins do I need to get from Iron 1 to Diamond?

88 wins

With a 55% win rate, you need to play 880 games, win 484 of them and lose 396.

With a 60% win rate, you need to play 440 games, win 264 of them and lose 176.

With a 65% win rate, you need to play 293 games, win 190 of them and lose 103.

With a 70% win rate, you need to play 220 games, win 154 of them and lose 66.

Assuming an average LP gain of +/-25 LP per game below Emerald and +/-20 LP for Emerald+ as per Riot Phroxzon's tweets here and here.