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Doubly mutilated chessboard problem: not the solution

On Friday I wrote about the mutilated chessboard problem.

I'll give the solution now. The reason this problem is usually phrased in terms of a "chessboard" is that a chessboard has its squares alternately colored white and black. (I'm going to call the colors white and black even if that's not actually what they are.) A chessboard has the same number of squares of each color, and any domino covers one white square and one black square. So if we remove two squares of the same color, such as the opposite corners, then tiling is impossible.

It turns out that if you remove any two squares of different color, leaving 31 squares of each color, the board can be tiled with dominoes. Drini gave the solution; draw any tour that goes through all the squares exactly once and finishes where it began. Removing two squares of opposite colors breaks this into two subpaths of even length, which are easily tiled.

As for the other problem I asked, when can you remove four squared and have there be a tiling: clearly we need to begin by removing two squares of each color. There is another necessary condition for there to be a tiling -- the squares removed can't block off a single corner square from the rest of the board. (You might worry that they could disconnect the board in some other way, but the only other way possible under the tiling constraints is that they disconnect a domino-shaped region.) But then the tour argument above doesn't work; in particular, a tour is broken up into four subpaths of even lengths if and only if the removed squares occur on it in the order black-white-black-white. Furthermore, it appears that there are ways to remove four squares from the four-by-four board such that the remaining twelve are tileable, but there is no tour that they break up into even-length subpaths. There aren't that many tours of the four-by-four board, so it's easy to check this by brute force with a simple program; in fact there are six according to Sloane's encyclopedia. (It says twelve there, but Sloane refers to directed tours.)

However, that cuts down the number of cases one has to deal with, and in the end it looks like a four-by-four chessboard, with four squares removed, is tileable if and only if the removed squares are of opposite color and do not include both of the squares adjacent to any corner.

My method (which I realize I haven't fully explained here; if anybody wants to seriously tackle this problem for some reason I'll be happy to explain in more detail) requires a bunch of nasty case analysis, though. And in the case of the 8-by-8 board there are millions of tours, so it doesn't generalize well.

121 comments:

Je said...

How about an intermediate problem: It's clear that if there is a tour in which the four removed squares alternate black/white then you can tile the complement. But is the converse true? Given a tiling of the complement, does it give you a tour in which the removed squares alternate black/white?
Or, one could try to solve the problem by showing that you can always modify a path where they don't alternate to produce one where they do.

Jesse Johnson said...

I hit enter at the wrong time while typing in my name. The above post is from me.

Sune Kristian Jakobsen said...

Interesting problem! I think I have proved that if you remove two squares of each color from a chessboard, without disconnecting a corner, you can cover the rest of the chessboard with dominoes. This cannot be improved to three squares of each color: If you remove a2, b3, and c2 (and three squares of the opposite color), there is no way to cover both a1 and b2 without covering b1 twice. It would be interesting to find out how many squares you could remove if the chessboard had more than 2 dimensions.

The proof:

We say that two squares are neighbors if they have a common edge. For a set of black squares B, let N(B) be the set of squares that are neighbor to a least one square in B. Now the idea is, if for all set B of not-removed black squares, the number of not-remove squares in N(B) is at least |B|, then there is a tiling, because of Hall’s theorem (http://en.wikipedia.org/wiki/Marriage_theorem). If B is the empty set it is trivially true. If B consist of a corner, the above is true, since |B|=1 and at least one of the neighbors is not removed (we are assuming that the corner is not disconnected). B cannot be of the form {all black squares except the neighbors of a corner (e.g. a2 and b1)}, because B may not contain removed squares, and not both neighbor of a corner may be removed. Finally, there must be at least two black squares not in B, because B don’t contain removed squares.

Now I want to prove that if B is not of the above forms, then |N(B)|>=|B|+2, because that would imply that the number of not-removed squares in N(B) is at least |B|. To see that |N(B)|>=|B|+2, cover the squares in B with horizontal dominoes such that each domino cover squares in row (a and b) or (c and d) or (e and f) or (g and h). The dominoes covers |B| black squares and |B| white squares, and the white square are all in N(B). To prove that |N(B)|>=|B|+2 I only have to prove that there are at least two white neighbors to the dominoes. The dominoes forms 4 columns (ab, cd, ef and gh), and for each column that contains at least 1 and not 8 dominoes, there must be at least one white neighbor to a domino in this column. Thus: If B is a counterexample, there is a most one such column. Similarly, if the dominoes were vertical, there would be at most one row with at least 1 and not 8 dominoes. If B is not of the above forms, this imply, that B is either one square (not in the corner), two squares in the same 2*2 square (e.g. a3 and b4) or all black squares except two in the same 2*2 square (and the two squares cannot be the neighbors to a corner). In all these cases it is easy to verify that |N(B)|>=|B|+2.

Sune Kristian Jakobsen said...

I forgot that you can remove both neighbor to a corner if you also remove the corner. This is not a problem: Because of symmetry we may assume the two neighbor are white. Now B can't contain the corner, and the proof works.

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I'm sorry to say that I have irrevocably failed to keep my addiction at bay and I have decided to watch some porn again. Do any of you guys know where I can gather up some quality porn pictures or porn:adult videos?? I'm specifically looking to download stuff without watermarks and sufficient quality dvd rips to build a nice and tidy collection. No tube crap. I've been searching for the past 6 hours to get hold of something like this. This is the first time I've visited www.blogger.com but so far I have only found a few videos and picz shared by the members. I got some great content from a guy calling himself ZefUnfissardceasp from another popular forum but my appetite is still not quenched. This is one of the videos he gave me just to give you an example of the quality I'm looking for:[url="http://pornvb.com/pro-porn-clips/70633-bridie-james-hottest-dogger-killergram.html"]bridie-james-hottest-dogger-killergram-pornvb porn forum[/url]

Anyway, I hope this is the right category to post this question in.


goodnight



Most people are only alive because it is illegal to shoot them.

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