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How many wins do I need to get from Silver 3 to Diamond?

64 wins

With a 55% win rate, you need to play 640 games, win 352 of them and lose 288.

With a 60% win rate, you need to play 320 games, win 192 of them and lose 128.

With a 65% win rate, you need to play 213 games, win 138 of them and lose 75.

With a 70% win rate, you need to play 160 games, win 112 of them and lose 48.

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.