Catch 21: Lies in Advertising

Question

I posted a few Dominosa puzzles this week to celebrate adding Dominosa to my Donimoes collection of domino puzzles and games. Now, I want to finish the week with something a little different.

What false claim is on this box? How do you know?

Answer 1

The lie is:

That there is only one logical solution (see the paragraph beneath the large 'h').

After all:

If you look at the 1-5 domino on the right-hand edge it is beside a space for a 5-5 domino. Since the spare pieces around the edge appear to fill all missing spaces (albeit with either the 2-4 or 3-4 out of shot and unseen) the 1-5 is in a valid position. BUT if you rotate this domino 90 degrees it will still fit into the solution, and still have a space beside it for the 5-5 - a second logically sound solution!

Extended answer: Furthermore, as @LannyStrack points out in comments:

The square of 1-3 and 3-3 dominoes in the top right corner also has two valid arrangements.

But there's still more!

We also have two valid arrangements for the square involving the 1-2 (not yet placed) and 2-5 dominoes at the bottom of the image and for the square left of centre comprising the 3-5 and 4-5 (not yet placed) dominoes.

In fact, if we play this game for real and begin placing dominoes on the empty board (noting that the unseen bottom left corner must be a 2 since all numbers should appear exactly 7 times) we end up able to place all dominoes in a unique configuration except for these 4 pairs of dominoes. This means there are 2^4 = 16 different solutions when you consider all permutations of these 4 ambiguities. One logical solution indeed!

Explanation of logic for deducing points in the spoiler block above:

It's initially tricky to find an 'in' to this puzzle, as each of the 21 dominoes appears in two or more places. The trick is to note that the 6-6 tile must lie on the left-hand side, and must involve the middle of the three 6's. This means it cannot be used as part of a 4-6 domino with the 4 to its right; this leaves only one possible position for the 4-6 domino (bottom row). Placing this domino automatically resolves both the 2-6 and the 6-6.



We can now uniquely place the 1-6 domino on the right-hand side of row 3. This then leaves just one legal position for the 5-6 (top, left of centre), which in turn leaves one legal position for the 3-6 (bottom right). Placing the 5-6 also subsequently forces the placement of the 3-4, 1-4 and 2-4 dominoes.



We can now place each of the 4-4 and 1-1 tiles in their only remaining legal position. Trying to do the same for the 2-2 domino reveals two potential positions for it (top of column 4 & middle of row 6). However, note that the row 6 option would divide the bottom unsolved section into two separate areas of size 5 and 7 - it would be impossible to fit dominoes in these perfectly (since by their nature we always require an even number). Therefore, the 2-2 goes in column 4. This now leaves one legal position for the 2-3 domino (row 6).



Finally then, we arrive at a position where no more logical deductions are possible, and we are left with the 4 pairs of ambiguous dominoes previously stated in this answer.