There are certain logical deductions that are not made in this solution. For instance, we know that the large square AB/GH does not contain the two dominoes 1-4 and 2-4 because we can assume that the solution to the problem is unique and there would be two ways of correctly arranging those two dominoes in that large square. (The same argument applies to BC/KL.) During the test, I did make these assumptions in the name of speed, and the interested reader can see how much faster the solution is with those few extra lines drawn in.
There are also a few logical deductions that are made without comment. For instance, when I identify that 2-5 is located in a specific place, I will also draw lines between all of the other adjacent 2's and 5's in the grid.
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The only place for 5-5 is AN/BN. The 2-2 must include BM, so BL/BM is not a domino. The 3-3 must include CL, so CL/CM and CL/DL are not dominoes. The 4-6 must include AJ, so AJ/BJ is not a domino. AG must be part of the 2-4 domino, so EM/FM is not a domino. |
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Let's assume that 2-2 is AM/BM. Then the only 2-5 is
EJ/FJ and the only 1-5 is EL/FL. Then EK/FK must be the 1-3.
The 3-5 must include EG, so AL/BL is not a domino. 3-3 must be at
BL/CL and 3-6 must be at CJ/CK. Therefore, 6-6 must be at CN/DN, but
then we would have 2-3 at both CM/DM and EM/EN. Therefore, AM/BM is
not a domino.
With that one line in place, we know that AL/AM, BM/CM, and CN/DN are dominoes. |
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3-4 and 3-2 are at MD and NE in some order, so 3-1 and 3-5
are at EG and FL in some order. Therefore, 3-3 is BL/CL and 3-6 is
CJ/CK. 2-6 is FH/FJ, so GE/GF is the 3-5. The 1-3 must be at
EK/FK.
DK/DL is not a domino, so that the lower-left section contains an even number of squares. DJ/DK must be the 1-6, EH/EJ is the 1-5, FL/FM is the 4-5, EN/FN is the 3-4. DM/EM must be the 3-2, so DL/EL is the 4-1 and the following are dominoes: AG/AH, AJ/AK, BJ/BK, BG/CG, BH/CH, and DG/DH. |
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Here is the completed grid with the horizontal dominoes in yellow. |
Notes: I got through this problem in about ten minutes during the test. Fortunately for me, I made the correct guess as to where the 2-2 domino was, so I didn't have to go through the second frame of logic during the test.
It has been pointed out that this problem is (accidentally) the same one that was asked in the 1999 qualifying exam. It's gratifying that I was able to solve it this year, although perhaps I held some subconscious memory of it.