Ken's POTW 010327

A Six-Color Square

Color each square of a 6x6 grid of squares with one of six colors (say A, B, C, D, E and F) such that:

Each row contains each of the six colors.
Each column contains each of the six colors.
Each main diagonal contains each of the six colors.
The squares along each main diagonal cannot touch squares of the same color, even at corners. (Same-color corner-touching is allowed off of the main diagonals.)

Source: A reader's physics teacher. (If anyone has a better source, let me know.)

Solutions were received from Colin Bown, Samantha Levin, Dan Ghinea, and Denis Borris. Dan Ghinea sent results showing there are 2880 ways to color such a square with six colors. Dividing by 6! (the number of ways to arrange 6 colors) leaves 4 solutions with ABCDEF on top:

Solution 1:
 A B C D E F      
 D F E C B A      
 C E B A F D      
 F A D E C B      
 B C A F D E      
 E D F B A C

Solution 2:
 A B C D E F  
 E D A F C B  
 C F B E A D  
 F E D C B A  
 D A E B F C  
 B C F A D E

Solution 3:
 A B C D E F  
 E D F A C B  
 C F E B A D  
 B C A F D E  
 F E D C B A  
 D A B E F C

Solution 4:
 A B C D E F
 F E D B A C
 C A F E B D
 E D B C F A
 B C A F D E
 D F E A C B

From solution 1:

Rotate left 90 degrees, replace ABCDEF with BDFCEA to get Solution 2.
Rotate right 90 degrees, replace ABCDEF with AECBFD to get Solution 3.
Rotate 180 degrees, replace ABCDEF with BCAEFD to get Solution 4.

So there is only one solution, after rotations/reflections/replacements.

(BTW, in solution 1, you can see a vertical symmetry between A/E, B/D and C/F. This same symmetry can be seen in the other solutions to more easily see their equivalence.)

Here is Denis Borris' logic (which led to two duplicates of the first solution above - rotate the second 180 degrees.)

Did it this way: 

1- entered 123456 in lefttop-rightbottom diagonal 
2- earmarked corresponding squares with their no-no's 
3- assumed a difference of 4 gave better odds at center 
of other diagonal; so entered 1,5 (then 2,6 for 2nd try) 
4- repeated step 2 
5- following that,everything "cancelled out" or "fell in" 
beautifully; as example, in 1st solution, only "1" was 
possible in the square at end of 2nd row** 

So here are my 2 solutions:

1 3 6 5 4 2 
5 2 4 6 3 1** 
6 4 3 1 2 5 
2 1 5 4 6 3 
3 6 1 2 5 4 
4 5 2 3 1 6

1 6 4 5 2 3 
3 2 5 6 1 4 
4 1 3 2 6 5 
2 5 6 4 3 1 
6 4 1 3 5 2 
5 3 2 1 4 6

Update 4/30/01: Hareendra Yalamanchili sent a logical approach to a solution:

Enter ABCDEF as top row.

Fill in diagonal starting at top left with first possible letter.
A B C D E F
  D        
    B       
      C     
        F   
          E 

Fill in second diagonal starting at bottom left with first possible
letter.

A B C D E F
  D     C   
    B E    
    D C     
  A     F   
B         E

Then the rest fill in easily, starting with E&B in 5th and 4th rows.

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