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

52 wins

With a 55% win rate, you would need to play 520 games, you would win 286 of them and lose 234.

With a 60% win rate, you would need to play 260 games, you would win 156 of them and lose 104.

With a 65% win rate, you would need to play 173 games, you would win 112 of them and lose 61.

With a 70% win rate, you would need to play 130 games, you would win 91 of them and lose 39.

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.