Fictional story:

As everybody knows I'm not good at making puzzles so I ordered a Build-your-own™ Dominosa kit!

This should be good for my Dominosa building skills, I thought to myself. And today, the package finally arrived! I rushed to the door and grabbed the package. As soon as I opened it, six pieces fell out, with a one-line instruction:

Arrange the six pieces (no rotating!) into a 8x9 grid and solve the Dominosa

Hmm... so it wasn't going to teach me how to make a Dominosa puzzle, it's a puzzle itself. Simple, I thought. Although I can't make good puzzles, I can easily solve them! I put the pieces together, and tried to solve the Dominosa, but failed. I tried again and failed yet again. Weird. After a few more half-hearted attempts, I decided to post it here to see if you can find an/the answer:

enter image description here

  • $\begingroup$ Reflection allowed? $\endgroup$
    – boboquack
    Mar 4 '17 at 4:30
  • $\begingroup$ Nah, they're solid pieces. $\endgroup$
    – Wen1now
    Mar 4 '17 at 7:49
  • $\begingroup$ Computers allowed? $\endgroup$
    – Gareth McCaughan
    Mar 7 '17 at 2:40
  • $\begingroup$ @GarethMcCaughan Nah, because then that kind of ruins the point right? $\endgroup$
    – Wen1now
    Mar 7 '17 at 3:54
  • $\begingroup$ Is each 3X4 a Dominosa? $\endgroup$
    – Moti
    Mar 7 '17 at 6:25

The solved dominosa looks like this

At the start there is only one way to place an 88 and 38, this leaves only one way to place the 33 and just two possibilities for the 44. After that we try to place the 47 domino. There is only one 7 with three 4s for this domino. If it would make a 2x2 square with the 44 domino, we get a contradiction.

This means the 47 domino has to go to the right which gives us the placement of another piece. Now there is only one way to place the 24, which gives us the placement for the 44 and 45.

This leaves us with only one way to place the 46 domino and connects another piece. Thanks to this we know where our connected pieces are placed in the big dominosa and have some corners that help us in finding many dominoes.

There is only one placement for the 78 domino left. This lets us connect all the pieces to a big dominosa. Thanks to the new corners we can fill in two more dominoes.

Now all that is left is placing the remaining dominoes according to the normal dominosa rules.

  • 1
    $\begingroup$ Oh, this is good! +1 $\endgroup$
    – Techidiot
    Mar 7 '17 at 16:06
  • $\begingroup$ This is much faster than the intended solution. I should've checked to make sure there weren't any unique ways to join 3-8s or anything. $\endgroup$
    – Wen1now
    Mar 7 '17 at 21:11
  • $\begingroup$ @Wen1now What was the intended solution? $\endgroup$
    – w l
    Mar 8 '17 at 7:35

Having being working on this one for too long now, I wanted to post the complete solution.

Here's how the final grid looks

enter image description here

The idea

I am not sure if there is any well defined logic for arranging the parts of the grids. I tried several combinations before coming across this one which led through the final solution.

Some failed arrangements

This is one example of a failed arrangement enter image description here

Upon getting the right arrangements these were the steps I followed

1. 8-3 is a unique pair

enter image description here

2. 3-3 is another unique pair

enter image description here

3. 8-8 is another unique pair

enter image description here

4. Now, we can easily get 6-6 at the dead end

enter image description here

5. We get 7-8 as another unique pair

enter image description here

6. Now, as we already have 7-8, we can knock off 2-7.

enter image description here

7. This is the most important step and the only hardest part I came across
We can mark certain areas like this one - reason being we are going to have a 3-5 pair in the corner so it can't appear anywhere else.
enter image description here

Also, we can mark similar areas like this

enter image description here

8. Now, we have two routes, taking the first will eventually fail -

enter image description here

9. So, we take the other way by making pairs like 1-7, 7-6, 3-7, 1-6, 7-7, 5-5

enter image description here

10. Now, we get another unique pair as 1-5

enter image description here

11. Which leads to some new pairs like 3-5, 2-3, 2-1, 2-2, 6-3, 2-5, 1-3, 2-8, 3-4, 1-1, 4-8, 4-5, 2-4

enter image description here

12. Now, we already have 1-7 and 1-6, so we get 1-4 in top right corner.
enter image description here

13. Which leads to new pairs like 5-8, 4-7.
enter image description here

14. Final logical step is that, we already have a 6-6 pair, so just knock off 6-5 in the right.

enter image description here

15. Now, continue filling the grid in the same way to get the final solution

enter image description here



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