Knight Tour Latin Square
Problem
Copyright 11-14-2005, Dan Thomasson
Solution
www.borderschess.org/KnightTour.htm
Knight Tour Latin Square Tessellation
In the Knight Tour Latin Square shown above, only one instance of each integer
from 1 to 8 are in each row and column. The previous four 16-move mini-knight
tours were combined to make the Latin Square. The knight moves for each tour are
numbered from 1 to 8, then 1 to 8 is repeated a second time to complete the tour.
After filling in the four tours shown above with the same color as their line color,
they can be combined into one piece that can be tessellated.
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
There are all sorts of fun math oddities that can be found within this Knight Tour
Latin Square. Notice in the diagram below on the left that each alternating
horizontal number pair adds up to 9. Also, if you take the alternating horizontal
number pairs as one number each (i.e. 2 and 7 as 27), they are multiples of 9 (i.e.
27 is 9x3).
Alternating groups of 4 numbers each are shown in the above Latin Square on the
right. Consider each group of 4 numbers as one number (i.e. 2, 7, 8, 1 as 2781).
Divide 2781 by 9 to get 309. Do the same thing to all the 4-digit numbers on the
left half of the square. Add up these 8 numbers to get 4444. Now do the same thing
to the right half of the square. Notice that they also add up to 4444.
2781 / 9 = 309
1836 / 9 = 204
8163 / 9 = 907
7218 / 9 = 802
4527 / 9 = 503
3654 / 9 = 406
6345 / 9 = 705
5472 / 9 = 608
4444
4563 / 9 = 507
7254 / 9 = 806
2745 / 9 = 305
5436 / 9 = 604
6381 / 9 = 709
1872 / 9 = 208
8127 / 9 = 903
3618 / 9 = 402
4444
When all the numbers are divided by 9, the results are 3-digit numbers with a zero
in the middle. Also, the digits on either side of the zero for the first eight 3-digit
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
results on the left side of the board, contain each of the digits from 2-9. The same
occurs for the 3-digit results on the right side of the board.
I found it interesting that 4444 is also the exact number of solutions to the
excellent "Calmplex puzzle" designed by Andrew Snowie in Canada.
Look at the results when doing the same dividing/adding process for each entire
row:
27814563 / 9 = 3090507
18367254 / 9 = 2040806
81632745 / 9 = 9070305
72185436 / 9 = 8020604
45276381 / 9 = 5030709
36541872 / 9 = 4060208
63458127 / 9 = 7050903
54723618 / 9 = 6080402
44444444
Let's see how we can make a new 8x8 Magic Latin Square from the previous eight
8-digit results. All the rows, columns, and main diagonals should each add up to
44.
1. Remove all the zeros in each number to get the following eight 4-digit
numbers.
3957
2486
9735
8264
5379
4628
7593
6842
2. Place these eight 4-digit numbers down the left side of an 8x8 grid square as
shown in the square below.
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
3. Reverse the order (vertically flip) of the eight 4-digit numbers and place
them down the right side of the same square as shown above.
4. Add up all the rows, columns, and main diagonals to get 44 for each one.
5. Subtract "1" from all the digits inside the square to get a new Magic
Constant of 36.
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
With the new 8x8 Magic Latin Square, we can make an 8x8x8 Magic Latin Cube
where even the main diagonals of all faces of the cube, and the major internal
diagonals equals 44. Also, all rows, columns, and pillars equal 44. Look at the
eight levels. One row is rotated up each time to create a new level.
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
There are several number groups based on the number 9, or multiples of 9, that can
be found in the original Knight Tour Latin Square. Here are a couple groups. See
what other groups you can find.
Groups Containing all digits from 1-8
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
Groups Containing 4 digits adding up to 18
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
Here is another Latin Cube I designed that is based on a Magic Square originating
from 3-d Knight Moves: KTMS.htm. All rows, columns, main diagonals, and
pillars each add up to 260.
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm
Copyright 11-14-2005, Dan Thomasson
www.borderschess.org/KnightTour.htm