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When you fire the soccer balls, it does damage. I know that the power depends on the ball's color, and I also know that the damage output is affected by the number of EVs in the Pokémon. However, is it only EVs and the color of the soccer balls that affect the damage, and if so, what is the formula that determines it? This has been bothering me for awhile now. Thanks in advance.

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I'm not sure if there's a formula, but it does depend on the ball type and EVs.
Not sure if this is what you're looking for, so I'm explaining in a comment.
Ball Type:
Black: Average shot and power
Yellow: Multiple shots after charging with smaller shot size and has the lowest power
Orange: larger shot size with the second lowest power
Green: slowest shot speed, but has the largest shot size and highest power
Blue: Higher shot speed, smaller shot size, and the second highest power
EVS:
HP: increases shot size, less points are lost when hit by an attack
Atk/Sp.Atk: Increases power of the Pokemon's shot
Def/Sp.Def: Guts meter can fills faster, and the Pokemon can block longer
Speed: Increases shot and movement speed
Source:https://m.bulbapedia.bulbagarden.net/wiki/Super_Training
That is about half.  The thing I was curious about is if you have x number of Attack EVs (since that boosts power) with a certain ball color = y amount of damage.  Hopefully that makes sense.

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