Radlams

There are five tetrominoes:

tetrominoes1.jpg

All but one of them decompose into two dominoes:

decompositions.jpg

For the T, no matter which domino you remove, you are left with a pair of monominoes:

tetra_t.jpg

There are 12 pentominoes:

pentominoes2.jpg

All but one of them decomposes as a domino and a triomino:

pent_decomps1.jpg

For the X, you are left with three monominoes no matter which domino you remove.

xdecomps.jpg

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 P be a polyomino of size six or more. Remove a domino, D, from P. It is not hard to see that P-D consists of at most 4 pieces.

pminusd.jpg

If any one of the pieces of P-D (call that piece Q) is any larger than a monomino, then \{Q,P-Q\} is a nontrivial decomposition of P. There are only two cases that this proof does not cover:

two_exceptions.jpg

Both of which have nontrivial decompositions:

nontrivial_6.jpg

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:

small_radlams.jpg

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:

small_radlam_decomps.jpg

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 n-omino is removed, there is an m'-omino left in the decomposition, where m'\le m

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.

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One Response to Radlams

  1. Pingback: The Graph type of a Polyomino « Synchronicity

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