From the Super Mario Wiki
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Attack Pieces are special puzzle piece items that were introduced in Mario & Luigi: Bowser's Inside Story and its 3DS remake, Mario & Luigi: Bowser's Inside Story + Bowser Jr.'s Journey and reappear in Mario & Luigi: Dream Team. They are usually found in singular areas throughout the game, and collecting ten of them in one area will grant Mario and Luigi with powerful Special / Bros. Attacks in Mario & Luigi: Bowser's Inside Story, Mario & Luigi: Bowser's Inside Story + Bowser Jr.'s Journey and Mario & Luigi Dream Team (in Real World locations). Luiginary Attacks are also granted by collecting all 10 Attack Pieces in the Dream World locations from Mario & Luigi: Dream Team.
Most Attack Pieces can be found in and collected by striking blocks that resemble them, though others can be obtained by different methods (usually from NPCs after completing certain objectives). At times, Mario and Luigi can get either one or multiple Attack Pieces at once.
Mario & Luigi: Bowser's Inside Story / Mario & Luigi: Bowser's Inside Story + Bowser Jr.'s Journey
Attack Pieces make their debut in Mario & Luigi: Bowser's Inside Story, where they grant Mario and Luigi with a Special/Bros. Attack once all ten are collected in a single area. They are first found in the Trash Pit inside Bowser's body, and are later found in areas both inside and outside Bowser's body (such as Pump Works and Toad Town). The locations of the Attack Pieces remain the same in Mario & Luigi: Bowser's Inside Story + Bowser Jr.'s Journey.
Mario & Luigi: Dream Team
Attack Pieces make a return in Mario & Luigi: Dream Team, but this time, they grant different types of special attacks. In the Real World, Attack Pieces grant a Bros. Attack to one of the brothers once all ten are collected. In the Dream World, they instead grant a Luiginary Attack to Mario. The brothers obtain their very first Attack Piece in the Real Word in Mushrise Park, given by a Toad after saving him from Thorbs, and in the Dream World in Dreamy Mushrise Park.