Dissections of pentominoes into tetrominoes
Livio Zucca considered the problem of dissecting a pentomino in the minimal number of pieces which can be rearranged to form a tetromino. Since there are 12 pentominoes and 5 tetrominoes we have 12 × 5 = 60 subproblems. Livio's solutions are depicted below (two have been improved by plinius, Silvio Sergio, and il Raccattapalle).
Livio considered also the problem of finding the minimal set of pieces that can be used to build every pentominoes and every tetrominoes (of like area). The solution proposed by Livio is the one here with 9 pieces and the construction of the various pentominoes and tetrominoes is depicted at the end of this page.
For other dissection problems see here. If you could improve some dissection let me know! (Almost all drawings by Livio Zucca).
As can be seen from the picture, every pentomino (apart pentomino X) can be seen as formed by the domino and one of the two triominoes. Thus it is sufficient to verify whether the pieces can cover the tetraminoes, the pentomino X and the domino plus each of the triominoes.