More Dominoes


As with the earlier problem "Domino Magic Square", this problem also asks for a magic square. From a regular set of dominoes, remove all blanks, leaving 1-1, 1-2, ..., 6-6, 21 dominoes in all. Remove three dominoes of your choice and arrange the remaining 18 dominoes in a 6x6 grid, such that the sum of the dots in each row, column, and major diagonal is the same.

For example, removing the 6-6, 5-6, and either of 4-6, 5-5, will result in each sum being 19. Removing 1-1, 1-2, and either of 1-3, 2-2, the sum must be 23. (Feel free to prove this.)

I usually do not post problems to which I have no solution, but I have made an exception here. I came within 2 dominoes of a solution, so I feel that one must exist. Please send whatever solutions you have. As always, I will post any unique solutions and credit the solver.

Source: Original.

Solution:
Wilfred Theunissen and Evert Offereins sent me literally hundreds of solutions to this problem. I asked them if they could find any solutions that were somehow unique and I would highlight them. They found solutions for the sum being 19, 20, and 21. For solutions to 22 and 23, the numbers x for 19 and 20 need merely be changed to (7-x). Here are some of the solutions they found:
For sum 19: 
+++++++++++++
+1 1+1+4+6+6+
+++++ + + + +
+6+6+2+1+1+3+
+ + +++++++++
+2+4+3 1+5+4+
+++++++++ + +
+5 2+5 4+1+2+
+++++++++++++
+2 3+4+5 3+2+
+++++ +++++ +
+3 3+4+4 3+2+
+++++++++++++

+++++++++++++
+1 1+1+4+6+6+
+++++ + + + +
+6+6+2+1+1+3+
+ + +++++++++
+2+4+2 4+5+2+
+++++++++ + +
+5 3+4 3+2+2+
+++++++++++++
+2+1 5+4 4+3+
+ +++++++++ +
+3+4 5+3 1+3+
+++++++++++++
For sum 20: 
+++++++++++++
+1 1+1+5+6+6+
+++++ + + + +
+6+6+2+1+1+4+
+ + +++++++++
+2+3+4 1+5+5+
+++++++++ + +
+3 2+4+6+3+2+
+++++ + +++++
+3 3+5+5+2 2+
+++++++++++++
+5 5+4 2+3 1+
+++++++++++++

+++++++++++++
+1 1+1+5+6+6+
+++++ + + + +
+6+6+2+1+1+4+
+ + +++++++++
+2+3+3 2+5 5+
+++++++++++++
+3+4+3 5+4 1+
+ + +++++++++
+4+4+5 2+3+2+
+++++++++ + +
+4 2+6 5+1+2+
+++++++++++++
For sum 21: 
+++++++++++++
+1 1+1+6+6+6+
+++++ + + + +
+6+6+2+1+2+4+
+ + +++++++++
+3+5+1 3+4+5+
+++++++++ + +
+4 5+5 5+1+1+
+++++++++++++
+4+2+6+2+5 2+
+ + + + +++++
+3+2+6+4+3 3+
+++++++++++++

  +++++++++++++
  +1 1+1+6+6+6+
  +++++ + + + +
  +5+5+2+3+2+4+
  + + +++++++++
  +4+5+4 2+3 3+
  +++++++++++++  <------ anti-symmetric for this line.
  +3+2+3 5+4 4+
  + + +++++++++
  +2+2+5+4+5+3+
  +++++ + + + +
  +6 6+6+1+1+1+
  +++++++++++++

  anti-symmetric solution: board[i][j]+board[7-i][j]=7 !!
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