The problem consists in taking two polyominoes, say a tetromino and a pentamino, and finding a figure which can be tessellated indipendently by both the polyominoes. Like these:

or
I discovered and became interested in this problem thanks to the great pages of
Livio Zucca (and in particular Page1, Page2, and Page3) and of Jorge Luis Mireles which collect a large number of results, mostly obtained by hand by a small group of polyomino "heroes".

I can't solve these problems by hand. (I have difficulties even in telling if two polyominoes are the same apart reflection and rotation...), so I wrote a little program that searches solutions for me, since I like to look at them.
In the following pages I collect the results I obtained running my little program.

I must point out that a large fraction of the solutions I report here has been probably already obtained by other people. I apologize for this, but I'm not really interested in the game of I did it before you did, nor I have the time to compare all the solutions and attribute ownership. I will do it only in the few cases in which the computer program does not find an already known solution.

Visit Col. George Sicherman web site for a large number of beautiful results on this and related problems.

I apologize for the names of the polyominoes. I did most of the work during the Christmas holidays, when I was out of Internet, so I did not have a model to conform to. After that, I was too lazy to change all the names...
A list of the polyominoes I have used is here.

Update Col. George Sicherman has found a lot (at least 100) of wonderful new solutions. These below are just 4 examples, but to contemplate his most amazing accomplishments please visit his Addendum page.

George Sicherman also discovered that in 2011 a Japanese programmer (nickname: deepgreen) did find a minimal solution for 5T=4Q:

 Pairs Web pages Solved Pentomino - Tetromino tetrominoes 58 / 60 Pentomino - Pentomino pentominoes 61 / 66 Hexomino - Tromino trominoes IL 70 / 70 Hexomino - Tetromino tetrominoes 166 / 175 Hexomino - Pentomino pentominoes 389 / 420 Hexomino - Hexomino hexominoes 550 / 595 Heptomino - Tromino trominoes IL 216 / 216 Heptomino - Tetromino tetrominoes 473 / 540 Heptomino - Pentomino Pentominoes FILN Pentominoes PTUV Pentominoes WXYZ 1102 / 1296 Octomino - Tromino tromino I tromino L 735 / 738 Octomino - Tetromino tetromino I tetromino L tetromino N tetromino Q tetromino T 308 / 369 369 / 369 361 / 369 144 / 369 361 / 369 Octomino - Pentomino pentominoes Enneomino - Tetromino tetromino I tetromino Q Dekamino - Tetromino tetromino L

Nov. 2013. A fantastic result from Col. George Sicherman. He finally found a shape compatible with 8 pentominos: