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The Mario Cap (or Mario's cap) is an important part of Mario's clothing, and one of the most iconic clothing items in gaming. The cap is red and has Mario's "M" symbol; he is almost never seen without it in his adventures.
When Mario collects almost any power-up, his cap will either disappear (as with the Frog, Tanooki, or Penguin Suits) or change appearance or color with the rest of his normal outfit (ex.: Fire Mario, Ice Mario, Rock Mario, Cloud Mario, etc.).
Mario was given his cap in his first appearance in Donkey Kong because of the limited advances in video game graphics when the game was made. A red cap was added because programmers found it difficult to create the hair movement that would occur when the character jumped.
Super Mario series
Super Mario 64 / Super Mario 64 DS
The cap has appeared with Mario ever since Donkey Kong, but was unimportant until Super Mario 64 - the first game in which Mario can lose his cap. If his cap is lost (having been stolen by a Ukiki or Klepto or blown off by a gust of wind, for instance), Mario takes more damage when attacked.
In the remake of Super Mario 64, Super Mario 64 DS, Toad reveals that the cap Mario wears is special, and that if it were to get blown off of Mario, he would have nothing but bad luck. As hinted in Super Mario 64 DS, Mario's Cap may be a source of his strength, though this does not apply in other games where Mario can have his cap off. If Mario loses his cap, he can get it back by going back to the course in which he lost it to find it. However, if he instead gets a Power Star from another course, he can simply talk to Toad (who will find and wear Mario's Cap) and get it back from him. Mario's Cap appears in the following courses, even when he is locked: Bob-omb Battlefield (Star 2 or 4), Whomp's Fortress (Star 1 or 2), Jolly Roger Bay, and Cool, Cool Mountain (Star 2). Mario's Cap can also be found in most levels when playing as Yoshi. Yoshi, Luigi, and Wario can pick up the cap, which gives them the appearance and abilities of Mario (though they retain their voices) until they lose it, get hurt, or finish a level.
Mario also makes use of three different types of caps in addition to his regular cap: the Wing Cap (which enables Mario to fly), the Metal Cap (which enables Mario to walk underwater and through fire), and the Vanish Cap (which enables Mario to walk through certain walls). While these powers are dispersed amongst Mario and his companions in Super Mario 64 DS, Mario's Cap gains a new ability: when worn, it turns Luigi, Yoshi, or Wario into Mario. If Mario loses his cap, he can't fly, use other caps, or use Power Flower abilities.
Due to certain glitches, it's possible for Mario to end up with more than one cap; the duplicate can be used as an effective weapon. In Super Mario 64 DS, if someone wearing Mario's Cap gets another cap, they get an extra life. However, if the real Mario gets another one of his caps from a certain glitch in Snowman's Land, he'll pick up the cap and nothing will happen.
Super Mario Sunshine
The same effects of Mario's Cap in Super Mario 64 hold true in Super Mario Sunshine, but with the added effect of Mario gradually losing health due to the heat. The only enemies that steal Mario's Cap in this game are Swipin' Stus.
Due to a glitch, Mario will still lose health while speaking to NPCs and can even perform his death animation. However, the "Too Bad" screen will not appear if this occurs, only showing up after Mario finishes talking to the NPC.
Super Mario Galaxy
In Super Mario Galaxy, Luma hides in the Mario Cap once he joins Mario. Every time the player performs a spin, there is a small waiting time before Luma goes back into the cap to perform the move again.
New Super Mario Bros. Wii
In New Super Mario Bros. Wii, Mario is capless when he starts a level with 99 lives, though his victory animation at the end of a level doesn't change; his cap reappears on his head should he lose a life again.
Super Mario Galaxy 2
In Super Mario Galaxy 2, the Mario Cap has the same role as in its predecessor; however, the cap's importance is expanded in the ending, when Luma takes it with him as a souvenir before returning to the Comet Observatory with Rosalina, and Mario appears shocked to no longer have it. Following this, Mario is seen capless throughout the credits, but regains his cap when he is playable again, presumably getting a spare cap. The same thing happens when the player is Luigi.
Super Mario 3D Land / Super Mario 3D World
In Super Mario 3D Land and Super Mario 3D World, Mario and Luigi lose their caps in their small forms. However, in the former game, once the player has 1110 lives, their regular forms lose their caps instead.
New Super Mario Bros. 2
In New Super Mario Bros. 2, similar to New Super Mario Bros. Wii, Mario is capless when he starts a level with 1110 lives.
New Super Luigi U
Super Mario Odyssey
In Super Mario Odyssey, the cap, among others, plays an exceptionally important role. During the initial battle on Bowser's airship, Bowser's final attack knocks the cap off of Mario's head and sends Mario flying off. With Mario gone, Bowser stomps on the cap and leaves it to drift into his airship's propellers, where it is completely shredded. A piece of the cap is retrieved by Cappy, who uses it to transform into an exact replica of the cap when he and Mario team up.
As shown in the group photo after obtaining 100% completion in the game, Mario's cap seems to have been restored to normal after the events of the game, as it is shown separate from Cappy in the photo.
In Luigi's Mansion, Mario's Hat is one of the five items that Mario lost, and Luigi must bring it to Madame Clairvoya to receive information regarding Mario's whereabouts. After searching through the mansion, Luigi eventually finds it in a washing machine from the Laundry Room (in the 3DS version's Hidden Mansion, it is instead found in the dishwasher in the Kitchen). When Luigi gives her the cap, she says: "Wh-what is this? If it isn't Mario's Hat! Why, it's so clean, as if... SOMEONE JUST WASHED IT! ... Sorry, sometimes I like to add a little drama, you know."
It appears Mario owns multiple caps, as Mario is also seen wearing one when he is trapped within his painting.
Super Smash Bros. series
In the Super Smash Bros. series, the color of Mario's Cap can be changed when the player chooses one of Mario's alternate costumes. Mario also adjusts his cap properly in one of his idle poses. In Super Smash Bros. Brawl and Super Smash Bros. for Nintendo 3DS / Wii U, Mario spins and takes his cap off, revealing his hair during his side taunt. In Super Smash Bros. Ultimate, Mario throws his cap and Cappy appears during his new side taunt. Cappy also has a chance of appearing in Mario's Super Jump Punch.
Mario Golf: World Tour / Mario Sports Superstars
Mario + Rabbids Kingdom Battle
Luigi's Mansion (Nintendo 3DS)
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