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

56 wins

With a 55% win rate, you would need to play 560 games, you would win 308 of them and lose 252.

With a 60% win rate, you would need to play 280 games, you would win 168 of them and lose 112.

With a 65% win rate, you would need to play 187 games, you would win 122 of them and lose 65.

With a 70% win rate, you would need to play 140 games, you would win 98 of them and lose 42.

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.