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 11, 12, ..., 66, 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 66, 56, and either of 46, 55, will
result in each sum being 19. Removing 11, 12, and either of
13, 22, 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 (7x). 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+
+++++++++++++ < antisymmetric for this line.
+3+2+3 5+4 4+
+ + +++++++++
+2+2+5+4+5+3+
+++++ + + + +
+6 6+6+1+1+1+
+++++++++++++
antisymmetric solution: board[i][j]+board[7i][j]=7 !!

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