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Whenever Mario enters water from a great height in Super Mario 64, he says "haha!" This actually sets a "haha flag", which does not get cleared until Mario enters an airborne state or exits the area. As long as it is set, Mario will say "haha!" every time he touches water.
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In Yoshi's Island, if Yoshi deals the final hit to Hookbill the Koopa before he hits the ground, Yoshi will shoot up into the sky at an extremely high velocity. The boss defeat cutscene will play out and the level will end without Yoshi ever coming down.
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Official Yoshi ringtone available for download from Nintendo of Europe's site in 2002. The ringtone is an arrangement of the Yoshi's Island track from Super Smash Bros. Melee.
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In Super Mario Sunshine, walking into the space between the slot machines in Casino Delfino will cause random parts of Mario's model to oscillate uncontrollably. He is actually walking extremely quickly and being displaced back on every frame; the rattling sound is his footsteps.
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It is widely known that the Endless Stairs in Super Mario 64 can be skipped by performing a series of backwards long jumps up the staircase. However, for some specific angles, Mario may get caught on an invisible wall and then warped down instead, entering Wet-Dry World.
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When the Nintendo logo appears at the start of Super Mario Sunshine, a coin sound is heard. The game is actually coded to sometimes have Mario randomly say "Ciao!" instead; however, this never occurs in practice due to a fault in the random number generator.
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@DDSeudonym If you mean this one, then yes! This comes from a 1996 manga. It is presented without context even within the work; Luigi only appears like this on the cover and not within the story itself. Thank you very much for your inquiry!