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King on the run (Posted on 2016-06-01) Difficulty: 3 of 5
A chess king starts at a position A in the top row of a standard chessboard. The number of paths of length 7 to a position B in the bottom row is a perfect square, but not a perfect cube.
The number of paths of length 7 from A to a position C in the bottom row is a perfect cube, but not a perfect square.
The number of paths of length 7 to a position D is both a perfect square and a perfect cube.

How many chess king paths of length 5 are there from B to C?

No Solution Yet Submitted by Ady TZIDON    
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Another way - 2 part Comment 5 of 5 |
Not to much easier but it is also possible to do this for the second part:

In an infinite grid there are 160 possible 5-lenght paths from a square to another in the same row at distance four. 

┬┐How you do know this? Because of the polinomial formula
https://www.quora.com/How-many-6-move-paths-can-a-chess-king-make-using-only-its-originally-adjacent-squares-ending-where-it-began
 
The simmetry implies that the number of paths going though squares in the row or in superior rows, is the same number of paths going under the row (no paths are just on the row in our case). 

Then the number of paths is 160/2=80?
No, because you have to sustract also the paths that are in part under and in part above the row. This are 20 (40 with simmetry). 

So 160/2-20=60 


  Posted by armando on 2016-06-03 05:40:31
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