Arrive in the Distant Spring on or before Day 15. Head south with blue pikmin. You should come across a large egg. Break it open and the Smokey Progg will come out. Lead it to the landing site and throw pikmin on its head. Keep throwing red, blue, or yellow pikmin on its head. Eventualy it will be defeated producing an egg thats worth 100 pikmin!
Once you have saved your progress once and you have all three onions, a new option will be available from the main menu: Challenge Mode. This mode lets you go back and compete in the areas that you have already opend up in the main game. The idea is not to find spaceship pieces, but rather to build as big a Pikmin army as you can before the sun sets. The biggest army scores are saved and entered into a ranking.
Alternately, complete one full day on The Impact Site. Get both parts while in the day. You will then unlock the "Challenge Mode" option when the Gamecube is restarted.
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When you have four Pikmin around you, hit down on the D-Pad and you will lay down. Four Pikmin will then carry you like anything else they carry to an Onion and when you got flying up into it, shortly after you land, there will be some fireworks from the Onion.