Re: dominos on a chessboard (was Re: multi-step reasoning)

["John S. Denker" <jsd@MONMOUTH.COM>]

Chuck Britton wrote:
> note that each domino MUST cover exactly one BLACK square and one RED square.

That's the key observation.

Note that you can dress this up in physics language:
It's a conservation law.  Let the red squares have positive
charge, while the black squares have negative charge.  The
initial configuration exhibits global charge neutrality.
The act of adding a domino conserves charge.  Therefore
(by induction) any final state must be neutral.

Jack Uretsky wrote:
>         Going to 3 dimensions and the possibility of inserting a domino
> edgewise, or some such trick, the answer would be different.

There are various ways of generalizing to D=3.  For the "normal"
generalizations, the conservation argument goes through unchanged.
You have to be slightly careful, because the principal diagonals
(in, say, the [1,1,1] direction) have alternating color, not constant
color, so there is nothing to prevent you from tiling the 8x8x8
"checkerboard" leaving opposite corners open, and in fact such
tilings are easy to construct.

OTOH if the assignment is to leave open opposite corners on one
_face_ (e.g. along the [1,1,0] facial-diagonal) then conservation
forbids it.