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Complete any stage by jumping only 30 times or less.

I did this in the City stage. If you strategically jump you end up jumping WAY less than 30 times. When I would jump I would use Aviator to get myself extra height and then dash until I had to land. For the first two screens I would only jump once and then dash to the end of the screen. On the third screen after killing the first enemy I would jump at the hill and dash until I got to the first green robot that shoots the spiked balls. I would kill him (I think it might take another jump, depending on how you land).

After he is dead I would jump again from his platform to dash past the second green robot and into the part where the floor drops. Once the floor drops I would just start air dashing. If using Aviator you can pretty much make it to the ladder. If you can make it to the ladder without jumping you can go through the waves of enemies without jumping. You gotta jump to clear the gap and I couldn't figure out any cool dash techniques to get past the next area in one jump.

Clear the gap and take out the first two enemies. On that first raised platform jump and you can dash clear to the next section.

Once you go under the wall you don't have to jump again until you reach the second set of green barrier enemies. You want to clear this gap in one jump. If you gotta take damage take damage but you wont have to jump again until the end of the hall.

Sadly you can't get to that ladder from jumping off the wall so this whole section takes two jumps up the wall, one jump to the ladder, and one jump off the ladder to the next section (4 jumps). From this section do not move off the raised platform. Kill the barrier enemy and then dash and air dash to the next one. At the end of this section its one final jump to the boss. Don't even bother jumping to avoid the boss, just take those hits.

In counting where I said to jump that is only 13 jumps. That gives you 17 extra jumps on the more tricky places or if you mess up.