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 Pushing Numbers 2 (Posted on 2007-05-22)
Here is the first part of Pushing Numbers.

You need to make it so that all the numbers from 1 to 8 appear once in every row and column. You can only do that by "pushing" the columns down and the rows left. Each time you "push" a row or a column all the numbers in it move one block in the direction you have pushed it.
 6 4 8 6 8 7 1 3 1 2 5 6 7 4 2 8 5 4 2 2 7 5 3 7 5 3 1 3 3 8 4 7 5 6 2 1 6 3 1 6 1 4 7 8 2 5 4 3 4 5 2 1 7 2 1 3 6 4 8 7 8 6 5 8
Can you do it?

 No Solution Yet Submitted by Assaf Rating: 3.5000 (2 votes)

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 re(2): No Subject Comment 6 of 6 |
(In reply to re: No Subject by Charlie)

Oops.  You are right!  I was tired when I wrote that, and it shows.  And, your insight about the number of vertical or horizontal moves in a row is correct--for each of those sequences of length n, we can divide the original 24! by n!.

I suspect the result will still be quite large, don't you?  Obviously much larger than my original estimate.

I think solving this will take a combination of a good computer program combined with some pruning heuristics to eliminate dead-end paths without searching them.  Knowing, for example that it's not easy to get a 4 into the left column, we could immediately stop searching move sequences in which getting a 4 there becomes impossible.  Similarly, it's not easy to get those four vertically consecutive 5s into separate columns--the program should elimnate paths where that's no longer possible.

 Posted by Ken Haley on 2007-05-29 00:51:48

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