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Solving the "Knight's Tour" math problem involves moving the knight about the chessboard such that each square is visited exactly once. Visualizing the chessboard as 4 quadrants, memorizing a small group of patterns within each quadrant, and following a few simple principles while calculating the knight moves will allow you to find a solution to this fun mathematical problem. It's an intuitive puzzle to challenge a friend, math teacher, or even a math classroom with.
Bishop's tour... DrFrank, I know you were joking, but perhaps you'd like to look at this link.
As for the Knight's tour, here's how I solve them:
Solving a Knight's Tour Blindfolded.
I had written a program in C to use recursion to solve this. At each try I would sort all 8 squares the knight could move to an ascending order (corner was the lowest followed by edge, center, and off-board ). Each square was visited in order (if on the board); if the square (member of 2-d array) had a zero, a move number was inserted there and it would "try" from that new square. If a "try" failed completely, control would return to the previous try, a zero would replace the move number on that square and it would attempt the next square on that try's ordered list.
There is a knight swap puzzle which might be of interest to those interested in a knight's tour: http://www.chess.com/forum/view/fun-with-chess/knight-swap-puzzle
The quadrant method is exactly the one I worked out some years back. A knight can't tour on a 4x4 board but the four-square patterns it can make on one build up into a tour when used on one quadrant after another.
Another method I worked out, earlier, is less pleasing to the eye but also works. A knight can tour on what's left of a 4x4 board after you remove two corners on the same edge (so the c3 - f3 - f6 - c6 zone of a chessboard, excluding c3 and f3 themselves). Then you need to begin a series of moves around the outer ranks and files (and c3 and f3) so that after completing exactly 25 of these (and leaving their mirror image untouched... that is, if you have visited a1 then you have left h1 alone, and so on) you can move to d3 or e3. After zipping around the centre zone, your last move on it is to e3 or d3 (whichever one you didn't visit first) and your final 25 moves are mirror images of your first 25.
This is much easier to do than to describe. The drawback of this method is that you can't make a re-entrant tour this way (that is, square 64 isn't a knight's move from square 1).
Alternatively, and this is easy to code up if you can program computers, just do the following:
DEFINE: Every square's "accessibility" is recorded, where this is the number of squares that are a knight's move from it. So a1 has accessibility 2, up to d4 which has accessibility 8.
EXECUTE: From the Knight's current square, select the square which is presently the least accessible of all those it can move to (provided it is accessible). In case of a tie, choose the one nearest the edge of the board. Having chosen, reduce the accessibility of all squares a knight's move away (because the square you are about to leave is no longer available) and move to the square you chose. Repeat until you have visited all 64 squares.
(With a slight modification I was able to use this program to tour a (2,3) leaper around a 10 x 10 board.)
Super! i really enjoyed this :)
Leonhard Euler (1707-1783) created a magic square where each rank and file totaled 260 and stopping halfway the sum is 130--of course a knight can reach each square in order (why else would I post it here?). http://www.johndcook.com/blog/2011/04/06/a-knights-magic-square/
nice video, well done!
Very simplified, thanks!
There are knight tour configurations that are magic squares. They are a real delight to view.
Cool. And yes, it would be very nice if you could make a video on the Bishop's tour.
Good One! Jerry; I have heard about this; but seeing this video for the first time; well done.
How about the Bishop's tour. Can anyone solve the Bishop's tour?
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