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When Mario or Luigi come in contact with a Fire Flower, they can become Fire Mario or Fire Luigi, respectively.
They gain the power to throw fireballs, red ones for Mario and green ones for Luigi. (These coloration differences have only existed in a few recent Mario games, though; both characters' fireballs are typically red or orange.) Their clothing also changes colors; their shirts and hats become white and Mario's overalls turn red while Luigi's turn green. This form was first introduced in Super Mario Bros. and was used in later side-scrolling games. The first 3-D game it appeared is Super Mario Galaxy.
In the Super Mario Bros. Super Show!, this form was called Super Mario or Super Luigi, while that usually is the name of the status when using a mushroom. The term "super" could just be tacked on to signify that they are stronger. Toad turned into this form in the episode "The Fire of Hercufleas". Princess Toadstool turned into this form in "The Trojan Koopa", although she didn't shoot fire.
Fire Power Without a Fire Flower
In certain games, Mario doesn't need a Fire Flower to emit flame from his hands. In Mario & Luigi: Superstar Saga, for instance, no Fire Flowers are involved, although Mario does learn to kindle a fireball in his palms from a mysterious character found in the Fire Palace. Mario is also seen shooting fireballs classic-style in his Fire Orb, Super Flame, and Ultra Flame attacks in Super Mario RPG: Legend of the Seven Stars, which only requires that he sacrifice a differing amount of Flower Points to do so. However, it should be noted that both of these mentioned games are RPGs; games of the RPG genre rarely involve traditional power-ups. Despite this, though, Mario and Luigi have been shooting fireballs without Fire Flowers in other games, such as Super Smash Bros. (although Mario gets an optional Fire Mario color scheme in Brawl) and Mario Kart: Double Dash!! (the latter of which actually features fireballs as the items that allow Mario and Luigi to shoot fireballs themselves), although most side-scrollers continue to use the Fire Mario and Fire Luigi forms and their typical manner of activation in a way unaltered from the original Super Mario Bros.