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.
Can you do it?

(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