Closed Knight's Tours

A Closed Knight's Tour means that the knight moves to all the 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."  Here is my thought process for creating a Closed Knight's Tour.
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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:

1. I make the same diamond pattern in all four quadrants using the directional pattern of the letter U.
2. I repeat step 1, but instead of diamonds, I make the same square pattern in all four quadrants.
3. I make the same diamond pattern in all four quadrants using the directional pattern of the letter C.
4. I repeat step 3, but instead of diamonds, I make the same square pattern in all four quadrants.

Select image for static Closed Knight's Tour solution.

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.

Closed Knight's Tour by Dan Thomasson, August 5, 2001

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!

16x16 Closed Knight's Tour by Dan Thomasson, September 13, 2001

I created the following 16x24 Closed Knight's Tour solution key which I used to solve the 16x16 Closed Knight's Tour, and it can be used on larger boards such as 32x32 squares, 64x64 squares, 104x104 squares, along with other board sizes.

16x24 Closed Knight's Tour Solution Key by Dan Thomasson, September 13, 2001 - Select image to see 32x32 Knight's Tour

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 tessellation patterns found in the Knight's Tour Tessellations section of this site.

Order 16 Magic Square Closed Knight's Tour by Dan Thomasson, February 18, 2002

The following image is a color coded number sequence of the Order 16 Magic Square Closed Knight's Tour shown above.

Order 16 Color Coded Magic Square by Dan Thomasson, February 18, 2002