Thursday, January 11, 2007

A Knight's Tour

A knight is placed on the centre square of a 5 x 5 'chessboard' (or 5 x 5 area of an 8 x 8 chessboard, if you prefer).

Using only the legal moves for a knight in the game of chess, is it possible for the knight to visit every square in the 5 x 5 area?

5 Comments:

Blogger steven said...

I should say that this puzzle has the condition that knight can visit each square once and only once, with the centre square as its starting point.

January 17, 2007 at 8:50 AM  
Blogger steven said...

Yes, it is possible. As with the water containers puzzle, one way to visualise this is with a graph with vertices representing squares and edges connecting each square to all the possible squares the knight can move to.

Starting from the centre, you can then draw a line connecting the vertices via the edges on the graph. It sounds complicated but it's simpler than it seems. It's largely a matter of visualisation.

More on knight's tours and graphs here :- http://mathworld.wolfram.com/KnightsTourGraph.html

One solution is as follows. Number each square 1-25, with square 1 being at the top left, square 25 at the bottom right. Square 13 is the centre square. The knight can follow the path 13-22-19-10-3-6-17-24-15-4-7-16-23-20-9-2-11-18-21-12-1-8-5-14-25. There are many other paths.

January 17, 2007 at 9:18 AM  
Anonymous Anonymous said...

No it is not possible. There are 13 squares of the same color as the center square (say "black") and 12 of the other color. Since every move of the knight consumes 1 black and 1 white, there will not be enough white squares to do it.

January 23, 2007 at 11:00 AM  
Anonymous Anonymous said...

Anonymous is wrong...sorry.

January 23, 2007 at 12:00 PM  
Anonymous Anonymous said...

Anonymous is wrong...sorry.

January 23, 2007 at 12:00 PM  

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