Super Mario Kart
Super Mario Kart
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Mario Part 4 - Did You Know Gaming? Feat. Yungtown
 
According to Nintendo, in a developer's interview, Donkey Kong Jr. was added because of both the 10th anniversary of his debut game, and that his shirt was easy to design.
Contributed by GamerBen144
In the Japanese release, a Boo sound is played when you unlock the Special Cup by pressing L, R, L, R, L, L, R, R, A on the Time Trial menu. However, you cannot unlock Special Cup with this method in the international version.
Contributed by MeleeWaluigi
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A prototype of the game has animations when ranking 2nd or 3rd place in a championship, that don't appear within the final game. Scoring 2nd shows a Super Koopa looping over the podium and then getting hit by the winner's champagne cork, whilst the 3rd place animation has a group of Koopa Paratroopas flying over the podium, the last of which is also hit by the cork.
Contributed by CuriousUserX90
Development of the game began with the idea of making a two-player racing game following F-Zero, which only had a single player mode. The game's prototype originally only had a generic "guy in overalls". The decision to include Mario characters was made roughly three to four months into development after the developers decided to add Mario driving one of the karts because they were curious about how the game would look. They ultimately decided the game looked better, and thus made the game a Mario series title.
Contributed by GamerBen144
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The official handbook actually encourages players to look at their opponents' screens, an action usually frowned upon in the gaming community.
Contributed by Boyobmas
A news story from the November 1992 issue of Daily Yomiuri (a part of Japan's largest media conglomerate) claimed that Luigi was actually evil:

"The most recent game, 'Mariokart', features races between Mario and his evil brother, 'Luigi', on a total of 24 Formula 1-style racing circuits..."
Contributed by Boyobmas
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The background used on the title screen in the Japanese version has what appears to be an Ai-Ai Gasa (or Love-Love Umbrella) saying "Mario" and "Peach", and "Mario Kart" written in Japanese. These were removed in the international version, and the mushroom was moved.
Contributed by ClaudX
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In the Japanese version, both Peach and Bowser's victory screen show them drinking the victory champagne, with Peach even having a red face, showing that she is drunk. This was changed to have Bowser just holding the bottle and Peach throwing it up in the air.
Contributed by gamemaster1991