By Gabriele Carelli
I apologise for my bed English, please let me know if you find some mistakes.
On february 5 2004 I proposed in the italian newsgroup it.hobby.enigmi a new a sub-class of polyominoes.
Let's draw the polyominoes marking all the sides of the squares and not only the external edge as usual; we say that a polyominoe is *Eulerian* if exist an Eulerian path in his graph.
Here are the Eulerian Polyominoes (EP) for N=1,....,7
It is not difficult, taken a polyominoe, to verify if it is Eulerian or not: we only need to count the number of vertexes in which converge only 3 segments, if they do not exist any or exist only 2 of them then the polyominoes is an Eulerian one.
Here are the EP for N=8:
Livio Zucca wrote a program to find all the EP untill N=12, here there are the serie achieved:
1,1,1,2,3,6,8,18,28,60,102,206,...
Here you can find the *.txt file with all the Livio's solutions.
Is not difficult to see that it's very hard use the EP to tile a rectangle while they easly tile the jagged rectangles, here there are 3 examples [ the first 2 are from Livio while the last one is mine]:
An open question is if exist an Eulerian N-minoe, with N>4, that tiles the plane.
Feel free to write me if you have something new about Eulerian Polyominoes.
Eulerian Polyominoes | Riding Polyominoes | Tetrapentominoes | Carelli's Puzzle| Squares_Pentominoes
Thi page is entrusted to the inexpert hands of Gabriele Carelli
First edition 27/03/2004 , last edition: 15/04/2004
If you have comments, questions or corrections feel free to write me