Closed (Re-entrant) Knight Tours
From the early 1940s to the late 1990s, GM George Koltanowski toured the countryside providing knight's tour
demonstrations at various chess clubs and tournament events.
Grandmaster George Koltanowski, 1903-2000
copyright(C) 2000, San Francisco Chronicle
He used a closed Knight's Tour for his solutions. "Closed" means that the knight moves
to all squares on the chessboard making legal knight moves and covering every
square only once in which the last move can connect back to the starting
position (64 connects back to 1). This is also known as "re-entrant". A member
of the audience would ask him to block out a square on an 8x8 chessboard. He
would begin making his knight moves from that square and then return to that
square with his last move.
Here is my thought process for creating a closed knight's tour.
When I create a tour, I think of diamond and square patterns and
the letters U and C.
I divide the board into four quadrants, start with the top
left quadrant, then place the diamonds and squares in their respective quadrants
following the directional patterns of the letters U and C.
Check out the following steps:
Step 1: I create the same diamond pattern in all four quadrants using the
directional pattern of the letter U.
Step 2: I repeat step 1, but instead of diamonds, I create the same square
pattern in all four quadrants.
Step 3: I create the same diamond pattern in all four quadrants using the
directional pattern of the letter C.
Step 4: I repeat step 3, but instead of diamonds, I create the same square
pattern in all four quadrants.
(click image to see elaborated version)
Look at the closed Knight's Tour below where I alternate the
diamond and square patterns for each quadrant. The result is the same - a
completed closed Knight's Tour. What once seemed impossible to do without
the aid of a computer is now possible and extremely simple.
(click image to see elaborated version)
The figure below shows a closed Knight's Tour on a 16x16 square board. The
pattern used in this example is the same pattern or solution that I use on
boards with 1024, 4096, or even 10816 squares. For speed and ease of
completing such large tours, it is best to use the same closed tour in the
16x16 board that I've provided, but begin the tour from the top left square.
In other words, make square 190 = 1, 191 = 2, 192 = 3, ... etc., all the way
around the board until the tour is completed. You will see just how easy it
really is!
I created the pattern in this solution key which I used to solve the 16x16
board Knight's Tour, and it can be used on larger boards such as 32x32
squares, 64x64 squares, 104x104 squares, along with other board sizes.
(Closed Knight's Tour Solution Key)
The following closed Knight's Tour has some very unique properties. Not only does
it make an interesting path pattern, but it is also an order 16 magic square where
all rows, columns, and main diagonals add up to 2056. The numerical sequence was
found in Madachy's Mathematical Recreations, by Joseph S. Madachy, Dover
Dubens, 1979, page 88. When drawing out the path pattern for each consecutive 16 moves,
notice the diamond and square patterns made in each quadrant. These patterns are
identical to the tessellational patterns found in the
Knight Tour Tessellations section of this site.
(Click image to see color-coded number sequence)
Enjoy the following two experimental closed knight tour sites where you can change the viewable size of the images from 25% to 1200%. Select the drop-down arrow at the bottom right corner of the screen to change the size from "Fit in Window" to other percentages. Unfortunately, you will need at least Internet Explorer version 5.0 or higher to use this feature.
Alternative Closed Knight Tours - I demonstrate three different types of alternative closed knight tours: 1) [(2,1) + (1,0)], 2) [(2,1) + (1,1)], and 3) [(2,1) + (1,0) + 1,1)]. (2,1) represents the normal move of a knight. (1,0) represents a vertical or horizontal move between two adjacent squares. (1,1) represents a diagonal move between two adjacent squares. The (1,1) move is similar to the move of the "fers" (a medieval chess piece also known as our present day queen piece). Combining the (1,0) + (1,1) moves would be the same as the moves of the king in modern chess. These alternative tours were created as a result of questions I received by GM Karsten Müller. He is now Dr. Karsten Müller after receiving his doctorate in mathematics. He wanted to know if it was possible to make closed tours using a combination of these alternative moves.
Closed Knight Tour Puzzles - Very entertaining geometric puzzles. When looking at the first six figures, notice the red border around figure (3) which represents the actual perimeter of figure (6). As you can see, figure (6) is only slightly larger than the original size of figure (3).
Take a look at a couple Centrosymmetric (180° Rotation-Symmetric) re-entrant tours by Walt Fristoe.
www.BordersChess.org/KTclosed.htm modified 2006.12.14
