There are five tetrominoes:
All but one of them decompose into two dominoes:
For the T, no matter which domino you remove, you are left with a pair of monominoes:
There are 12 pentominoes:
All but one of them decomposes as a domino and a triomino:
For the X, you are left with three monominoes no matter which domino you remove.
It is not hard to see that any polyomino of size six or more can be decomposed into two polyominoes neither of which is a monomino (let’s call such a decomposition a nontrivial decomposition). Here is one way to see that: Let be a polyomino of size six or more. Remove a domino, , from . It is not hard to see that consists of at most 4 pieces.
If any one of the pieces of (call that piece ) is any larger than a monomino, then is a nontrivial decomposition of . There are only two cases that this proof does not cover:
Both of which have nontrivial decompositions:
So if we wanted to study polyominoes with only trivial decompositions (which is what I had in mind originally) we are done already. But notice that these polyominoes:
have something in common. If you Remove Any Domino from any of them you Leave A Monomino (at least one monomino) in the resulting decomposition:
I like to call these RADLAMs (well, actually I like to write radlams) which is an acronym for Remove Any Domino, Leave A Monomino. A better name though is probably R2L1, because the definition generalizes:
Definition: An RnLm is a polyomino for which if any -omino is removed, there is an -omino left in the decomposition, where
For instance, we have seen R2L1’s above. Here are some R3L1’s:
Can you think of others?
I’ll have more to say about radlams, or R2L1’s in a future post. Until then, see what you can do with them, and let us know! I haven’t thought much about RnLm’s with n or m large, so even some nice examples of those would be cool.