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 Nov.Dec. 2010. Col. George Sicherman has found
these wonderful new solutions for the heptominopentomino (subproblems AVP and AVL), the octominotetramino (sub. LZN), the octominotromino (sub. MYI), plus some better (smaller) solutions for other subproblems.
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 