Re: covering a grid with dominos

In article <a79ee1pg7dg8aee610nk099lr8e47dqgk2@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:

> By a grid we will mean, as above, an m x n rectangular grid with a
> specified subset of cells removed.
>
> A cell can be identified by its cell address so cell (i,j) refers to
> the cell at row i, col j.
>
> Call the cell (i,j) "even" or "odd" according as i + j is even or odd.
>
> A grid will be called block connected if from any cell there is a
> "block path" to any other cell, where by a block path we mean a
> sequence of 1x1 cells in the grid such that successive cells in the
> path share a common edge. So for example, a 2 x 2 grid with the cells
> on the main diagonal removed is not block connected. For the warm up
> problems, all the grids were block connected except the grid for
> problem 2.
>
> It's easy to show:
>
> Given a block connected grid, if the grid can be covered by dominos
> then the number of even cells must equal the number of odd cells.
>
> I conjecture the converse:
>
> If a block connected grid has an equal number of even and odd cells
> then it can be covered by dominos.
>
> I believe the conjecture is true, but I haven't really tried the
> problem.
>
> Perhaps it's not true and there's a simple counterexample, or maybe
> it's true but well known -- I really don't know.
>
> In any case, I thought I'd pose it as a problem to the group.

xxox
oooo
xoox
xxox

This is a 4 x 4 grid with the cells marked x removed.
The remaining cells, marked o, are 8 in number, 4 even, 4 odd.
They are block-connected.
It's easy to see that they can't be covered by dominos.

--
Gerry Myerson (gerry@xxxxxxxxxxxxxxx) (i -> u for email)
.

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