From 3d Knight's Moves to Semi-Magic Squares and Cubes
What do 3-d Knight's moves have to do with semi-magic squares and cubes? We know that no 8x8 magic square exists for Knight's Tours, yet there are 140 Semi-Magic Knight's Tours of which 63 are closed tours and 77 are open tours as confirmed by MagicTour.free.fr. You may see my re-illustrated graphics of all the Closed Semi-Magic Knight's Tours at Knight's Tours Categories.
In the process of trying to make an 8x8x8 Magic Knight's Tour, I got stuck on the 50th move on the right side of the board of Level 3 without having any valid legal Knight's moves to continue the tour. Instead of giving up and scrapping the whole idea, I looked at the moves that were already made on the eight levels and noticed that they were making cubic shapes. To continue from move 50 to 51, and to finish making cubic figures in each level, I modified the move of the Knight to allow it to continue on the other side of the next level in the move sequence.
Though my efforts in making an 8x8x8 Magic Knight's Tour on this go-round have ended, I would like to show you how the cubic figures originating from the Knight's moves were instrumental in making a very nice 8x8 Semi-Magic Square.
Let me take you through my processes for trying to make an 8x8x8 Magic Knight's Tour. By sharing my ideas, perhaps you will become inspired to create your own Knight's Tours, Magic Squares, or Magic Cubes, and maybe even a Magic or Semi-Magic 8x8x8 Knight's Tour Cube.
Before embarking on any seemingly complex problem, it is best to write down a set of rules or guidelines so that you stick to a direct process without deviating. When you come to a point where the rules or guidelines no longer work, only then should you modify them or make up new ones. Here are my initial rules for trying to make the 8x8x8 tour:
Now for the fun stuff. I'll share some graphics that illustrate the above rules. Remember that at move 50, I had to modify the rules because there were no valid moves. Also, I cannot claim to make an 8x8x8 Knight's Tour since I modified the actual move of the Knight. I drew the following graphics in Microsoft Visio.
In the process of trying to make an 8x8x8 Magic Knight's Tour, I got stuck on the 50th move on the right side of the board of Level 3 without having any valid legal Knight's moves to continue the tour. Instead of giving up and scrapping the whole idea, I looked at the moves that were already made on the eight levels and noticed that they were making cubic shapes. To continue from move 50 to 51, and to finish making cubic figures in each level, I modified the move of the Knight to allow it to continue on the other side of the next level in the move sequence.
Though my efforts in making an 8x8x8 Magic Knight's Tour on this go-round have ended, I would like to show you how the cubic figures originating from the Knight's moves were instrumental in making a very nice 8x8 Semi-Magic Square.
Let me take you through my processes for trying to make an 8x8x8 Magic Knight's Tour. By sharing my ideas, perhaps you will become inspired to create your own Knight's Tours, Magic Squares, or Magic Cubes, and maybe even a Magic or Semi-Magic 8x8x8 Knight's Tour Cube.
Before embarking on any seemingly complex problem, it is best to write down a set of rules or guidelines so that you stick to a direct process without deviating. When you come to a point where the rules or guidelines no longer work, only then should you modify them or make up new ones. Here are my initial rules for trying to make the 8x8x8 tour:
- Draw eight chessboards and label them Level (or Plane) 1 through 8 on 8.5 x 11 graph paper. Place four boards in the top half of the paper going from left to right, then place the remaining four boards just underneath the first four. I find it easiest to use regular quarter inch graph paper in landscape orientation (long side of paper horizontal). It may be better to draw these boards in a computer program such as Visio, Paintshop, or Excel, then print several copies. This way you save time not having to redraw them yourself.
- Start with the Knight in the lower left corner square of Level 1.
- Move the Knight through the levels in the following sequence: 1-3-5-7-8-6-4-2-1 (Guenter Stertenbrink's suggestion).
- When moving the Knight from Level 1 to Level 7, first move the Knight directly up two levels (two squares) then to the right one square. If that move is not possible (because the Knight has already previously moved to that square or the move would be off the board), move one square left, else try moving up one square, then down one square if all other moves are not possible. Continue this process until the Knight reaches Level 7.
- When moving the Knight from Level 7 to Level 8, first move the Knight directly up one level (one square) then up two squares staying on Level 8, else down two squares if the first choice is not an option. If both first options are not valid, move right two squares, else left two squares if all other moves are not possible.
- When moving the Knight from Level 8 to Level 2, first move the Knight directly down two levels (two squares) then to the left one square. If that move is not possible (because the Knight has already previously moved to that square or the move would be off the board), move one square right, else try moving up one square, then down one square if all other moves are not possible. Continue this process until the Knight reaches Level 2.
- When moving the Knight from Level 2 to Level 1, first move the Knight directly down one level (one square) then up two squares staying on Level 1, else down two squares if the first choice is not an option. If both first options are not valid, move left two squares, else right two squares if all other moves are not possible.
Now for the fun stuff. I'll share some graphics that illustrate the above rules. Remember that at move 50, I had to modify the rules because there were no valid moves. Also, I cannot claim to make an 8x8x8 Knight's Tour since I modified the actual move of the Knight. I drew the following graphics in Microsoft Visio.
Level-1 - Knight's Tour Cube Construction
Level-2 - Knight's Tour Cube Construction
Level-3 - Knight's Tour Cube Construction
Level-4 - Knight's Tour Cube Construction
Level-5 - Knight's Tour Cube Construction
Level-6 - Knight's Tour Cube Construction
Level-7 - Knight's Tour Cube Construction
Level-8 - Knight's Tour Cube Construction
By using the cubic patterns (sometimes rotated or flipped) from Level-1 and Level-2 of the 3-d Knight moves, I was able to make a Semi-Magic Square where all the rows and columns added up to 260, but both diagonals added to 268. I reduced the size of the cubic shapes to fill single quadrants. You can see examples of both larger and smaller cubic patterns below. Each letter represents a number that is incremented by 8 from letter to letter (i.e. a=1, b=9, c=17, ...).
Large Cubic Pattern-1
Large Cubic Pattern-2
Small Cubic Pattern-1
Small Cubic Pattern-2
You can make 8 different cubes by connecting the like-colored numbers with lines. Also, notice that the like-colored numbers in columns are 8 numbers different (i.e. 9 - 1 = 8), while those in rows are 32 numbers different (i.e. 41 - 9 = 32). In the graphic below, I flipped or rotated a couple of the cubic shapes to provide the correct orientation for obtaining the total of 260 for rows and columns, and 268 for both main diagonals.
Semi-Magic Square derived from 3d Knight Moves by Dan Thomasson, November 25, 2005
Here are the results of my first 8x8x8 Magic Knight's Tour Cube. Though I've already discounted the ability to use the cubic shapes to make a true 3-d Knight's Tour Cube, I wondered if it might be possible to make a Magic Cube out of the same cubic Knight's moves. Each row, column, and main diagonal adds up to 2052 for every level of the 8x8x8 Magic Cube. Unfortunately, the pillars do not add up to 2052. Therefore, the following 8x8x8 Magic Cube is not 100% fully magic.
Level-1 8x8x8 Latin Cube by Dan Thomasson
Level-2 8x8x8 Latin Cube by Dan Thomasson
Level-3 8x8x8 Latin Cube by Dan Thomasson
Level-4 8x8x8 Latin Cube by Dan Thomasson
Level-5 8x8x8 Latin Cube by Dan Thomasson
Level-6 8x8x8 Latin Cube by Dan Thomasson
Level-7 8x8x8 Latin Cube by Dan Thomasson
Level-8 8x8x8 Latin Cube by Dan Thomasson
Take a look at the end of Knight's Tour Latin Squares II or the end of Knight's Tour Latin Squares III to see my Latin Cubes based on the previous 3-d cube. All rows, columns, main diagonals, and pillars each add up to 260, while each number is used only once per row, column, pillar (pile), or level (plane).
To see a nice modular 8x8x8 Closed Knight's Tour I designed on 11-19-2005, check out Knight's Tour Cube, and for an 8x8x8 Open Knight's Tour, see M-O-D-U-L-A-R Knight's Tours.
To see a nice modular 8x8x8 Closed Knight's Tour I designed on 11-19-2005, check out Knight's Tour Cube, and for an 8x8x8 Open Knight's Tour, see M-O-D-U-L-A-R Knight's Tours.