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:
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 heptomino-pentomino (subproblems AV-P and AV-L), the octomino-tetramino (sub. LZ-N), the octomino-tromino (sub. MY-I), plus some better (smaller) solutions for other subproblems.
|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||
|1102 / 1296|
|Octomino - Tromino||tromino I
|735 / 738|
|Octomino - Tetromino||
308 / 369
369 / 369
361 / 369
144 / 369
361 / 369
|Octomino - Pentomino||pentominoes||Enneomino - Tetromino||tetromino I
|Dekamino - Tetromino||tetromino L|