From the Super Mario Wiki
Red Coins are coins that sometimes appear along with Yellow Coins in the Mario series. The differences between a yellow coin and a red coin differ by game, but usually, red coins serve as important items to collect to get a reward. They are replaced by Purple Coins in the Super Mario Galaxy games, which must be collected to acquire Power Stars.
Super Mario series
Super Mario World
Super Mario World was originally going to have flying Red Coins that are worth five regular coins if collected, but they were unused during the game's development. They can be only accessed by hacking the game using a Super Mario World level editor.
Super Mario 64
Red Coins appeared in Super Mario 64, and returned in its remake. Each Red Coin is worth two Yellow Coins. Collecting all eight Red Coins in one level makes a Power Star appear over a Star Marker. In Super Mario 64 DS, Some Bob-omb Buddies have the ability to detect red coins when the player talks to them.
Red Coin Missions
Super Mario Bros. Deluxe
Red Coins are once again seen in Super Mario Bros. Deluxe. In the game's Challenge Mode, they (along with Yoshi Eggs) can be collected to obtain medals. Each stage contains 5 Red Coins, and collecting them all will earn the player that stage's Red Coin Medal if they successfully reach the end.
Super Mario Sunshine
Red Coins make another appearance in Super Mario Sunshine. Similar to Super Mario 64, collecting all eight in a level would make a Shine Sprite appear. However, Red Coins do not affect the player's total amount of Yellow Coins and are treated as a completely different currency altogether. Like Blue Coins, Red Coins replenish 2 health points on Mario's Life Meter.
Red Coin Episodes
Red Coin Secret Levels
In addition, red switches are located at the starting platform of Secret Levels, and when the player Ground Pounds the red switches, eight Red Coins appear and the timer is activated. Mario must then collect all 8 Red Coins before time runs out or he loses a life and must restart the level.
New Super Mario Bros.
In New Super Mario Bros., there are Red Rings in several levels, and if Mario or Luigi walks, jumps or swims through one, eight Red Coins appear. If he collects them all in the time limit, he is rewarded with a power-up. If Mario or Luigi are already using a power-up worth more than a Mushroom (i.e. a Fire Flower), he gets a 1-Up Mushroom instead of a power-up. Each Red Coin also increases the player's coin count by one, unlike Super Mario 64 in which each red Coin was worth two regular coins.
New Super Mario Bros. Wii
Red Coins and Red Rings also make a return in New Super Mario Bros. Wii. They serve the exact same function as in the preceding game and they will still give the player a power up if all coins are collected. However, small Mario collecting the coins would make a Fire Flower (or other power-ups of a similar power) instead of a Mushroom. Like before, if the player had already gotten a Fire Flower or similar power-up, they would get a 1-Up Mushroom after collecting all the Red Coins.
Super Mario 3D Land
Red Coins later appear with Red Rings in Super Mario 3D Land. Here, Mario must go through the Red Ring, then collect the Red Coins that appear before they disappear, like in other Mario side-scrolling games. Unlike other games, however, Mario must now only collect 5 of these coins.
New Super Mario Bros. 2
Red Coins appear once again in New Super Mario Bros. 2. Like in the previous New Super Mario Bros. games, Mario/Luigi must collect eight of them to earn a power-up, a 1-Up Mushroom, or a Golden Mushroom (only in Coin Rush mode), depending on his current form. When Mario/Luigi touches a Red Ring, Red Coins could appear in predetermined locations, or fall from the sky in parachutes.
New Super Mario Bros. U
Red Coins, along with Red Rings, reappear in the Wii U title New Super Mario Bros. U. Once again, all eight Red Coins must be collected for an item to appear. The similar Green Coins and Green Rings make their debut appearance, which are used in a similar way but five (technically fifteen) must be collected instead of eight. Green Coins also disappear quicker.
Super Mario 3D World
Red Coins and Red Rings reappear in the Wii U game Super Mario 3D World. They function as they did Super Mario 3D Land, awarding the next tier power-up when collected (depending on the level) and the highest tier power-up when another power-up is active. Rather than appearing in groups of five as they did in its predecessor, the coins appear in groups of eight as they traditionally do in the New Super Mario Bros. series.
Super Mario World 2: Yoshi's Island / Yoshi's Island: Super Mario Advance 3
Red Coins are first seen in Super Mario World 2: Yoshi's Island, in which they are hidden behind regular Gold Coins. Red coins have a slightly red tint to them, so they just barely stick out as different amongst gold coins and some gold coins are also tinted in red. When alone, neither feature is dramatic enough to be unambiguous. The only way to tell for sure which Gold Coins hide Red Coins is to grab them or use a Magnifying lens. Though visually the same size as gold coins, red coins are actually slightly larger, so that they are easier to grab. There are 20 Red Coins in each level, and grabbing one adds one point to Yoshi's score tally at the end of the level.
Yoshi's Island DS
In the sequel to Yoshi's Island, Yoshi's Island DS, Red Coins work the same way as they did in their first appearance; however, the Magnifying Lens was absent in that game. If the player can collect all the red coins in the level, the player gets a better score upon completing the level.
Mario Party series
Mario Party DS
In Mario Party DS, Red Coins replace Coin Bags in bonus mini-games unlike the previous versions of the Mario Party games. They are, like Coin Bags, worth five Coins. They are obtained like Coin Bags too; rarer and sometimes harder to acquire.
Mario Hoops 3-on-3
Red Coins appear in Mario Hoops 3-on-3 under the name 10 Coins. They can be found in ? Panels, and as evident by their name are worth ten normal Coins. They are only found if the player manages to combo the dribbling of acquiring the coins without being interrupted.
Luigi's Mansion: Dark Moon
Red Coins appear in the 3DS title, Luigi's Mansion: Dark Moon, with a slightly more octagonal shape. In some missions, there are hidden challenges where Luigi must collect the eight Red Coins within a time limit to reveal a treasure chest. They also appear as upgrade bonuses in the multiplayer mode, ScareScraper. Each time a floor is cleared, four Red Coins appear. Whoever collects the most Red Coins has a better chance of getting an upgrade.
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