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How many wins do I need to get from Silver 4 to Diamond?

68 wins

With a 55% win rate, you would need to play 680 games, you would win 374 of them and lose 306.

With a 60% win rate, you would need to play 340 games, you would win 204 of them and lose 136.

With a 65% win rate, you would need to play 227 games, you would win 148 of them and lose 79.

With a 70% win rate, you would need to play 170 games, you would win 119 of them and lose 51.

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.