From the Super Mario Wiki
Super Mario series
Super Mario Bros. 3
Their first appearance is in Super Mario Bros. 3. In this game, a Bull's-Eye Bill turns around once if it misses its target, but continues going forward without turning around after missing twice.
Super Mario Sunshine
Bull's-Eye Bills make another appearance in Super Mario Sunshine at Pinna Park's beach. In this appearance, they are purple and shot from the Monty Mole's cannon. They always home in, so there is no way Mario can easily outrun them. They are required to break open the baskets on the beach which hold Blue Coins. Bull's-Eye Bills explode when they make contact with an object, or if Mario squirts water at them. There is a gold version that will make coins appear if Mario jumps on it. During the final battle, Bowser Jr. sends some regular ones and also a blue one that will let out a 1-Up Mushroom or a Water Bottle if stomped on.
New Super Mario Bros. Wii
Bull's-Eye Bills reappear in New Super Mario Bros. Wii. They resemble normal Bullet Bills, but when one sees Mario, it will make a beeping noise and start flashing red. Here, they have a similar attack pattern to the one they did in Super Mario Bros. 3. However, in this game, they can change the angle they are flying at and will not stop pursuing the player until they are defeated or the player has gotten far enough away. They can be defeated the same way as normal Bullet Bills. They only appear in World 9-3, alongside Mario-seeking Banzai Bills.
New Super Mario Bros. U
In New Super Mario Bros. U, Bull's-Eye Bills make a comeback, appearing only during the Roy Koopa battle at the Rock-Candy Mines. They are seen being shot out of Roy's Bullet Bill cannon. In this case though, after some time of unsuccessfully chasing Mario, it self-destructs, blowing up like a Fire Chomp.
Super Mario Maker
Bull's-Eye Bills reappear in Super Mario Maker. As they did not appear in the original Super Mario Bros. and Super Mario World, new sprites were created for those two games. While they do operate similar to their appearances in New Super Mario Bros. Wii, the Bull's-Eye Bills in this game don't turn as sharply, and they'll only chase after Mario for a limited time period, instead of relentlessly pursuing him.
Mario Party series
Mario Party 8
Mario Party 8 introduced red Bullet Bills that do not home on the player. In the third round of the minigame Superstar Showdown, Bowser's Koopa Clown Car "spits" them. They move much faster than the ordinary Bullet Bills in the previous round.
Mario Party 9
Bull's-Eye Bills make a comeback in Mario Party 9 in the minigame, Billistics, where they appear alongside Bullet Bills. In this game, Bull's-Eye Bills make beeping radar noises. This is a way to tell if a Bull's-Eye Bill is coming. Like in other appearances, Bull's-Eye Bills will home in on the player, but when a Bull's-Eye Bill misses, it will not turn around, similar to Super Mario Bros. 3 when a Bull's-Eye Bill misses twice.
Captain Toad: Treasure Tracker
Bull's-Eye Bills reappear Captain Toad: Treasure Tracker. Unlike standard Bullet Bills, Bull's-Eye Bills are fired from red blasters. They behave like in past games and similarly to the Cat Bullet Bills in Super Mario 3D World, homing in on Captain Toad or Toadette and chasing them.
Yoshi's Woolly World
Bull's-Eye Bills make their debut in the Yoshi franchise in Yoshi's Woolly World, essentially replacing the Biting Bullet Bills from Yoshi's Island: Super Mario Advance 3. They are made out of red wool and are launched from red Bill Blasters. They can be found in World 6-4: Knot-Wing the Koopa's Sky Fort in the second half of the level and during the boss fight with Knot-Wing the Koopa himself.
Super Mario Run
Bull's-Eye Bills appear in Super Mario Run in Bullet Bill Barage. Bull's-Eye Bills attack from behind Mario, and they attack by making themselves level with Mario, then quickly moving forward to hit Mario.
Super Mario Run
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