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Draw the boundaries of the eight dominoes Question
Eight different dominoes lie on a plane. No boundaries between the dominoes are shown in the figure. Draw these boundaries.
Clarifications:
A domino is made up two square halves joined at a common edge.
In the figure above you see 16 domino halves. A domino consists of two horizontally or vertically adjacent halves.
Each domino half has zero or more dots on it. These dots are called pips.
For the purposes of this question, there is no restriction on the number of pips on the halves of one domino. For each domino, the number of pips on its two halves might be the same or different.
The question states that all domino are different. This means that if one of the dominos contains $M$ pips in one half and $N$ pips in the other half, there can’t be another domino with $M$ pips AND $N$ pips on its two halves.
The boundaries to be drawn are the outlines of each domino.
Attribution:
Quantum Magazine, November/December 1999
Art by Pavel Chernusky
2 answers
+---+---+---+---+
| |
| 0 0 1 3 |
+ +
| |
| 1 1 0 3 |
+ +
| |
| 2 3 3 2 |
+ +
| |
| 3 2 2 1 |
+---+---+---+---+
Step 1
The 3 in the bottom-left must be with one of the 2s next to it, so we can separate every other 2-3 boundary:
+---+---+---+---+
| |
| 0 0 1 3 |
+ +
| |
| 1 1 0 3 |
+ +---+
| | |
| 2 3 3 | 2 |
+ +---+ +
| |
| 3 2 2 1 |
+---+---+---+---+
Step 2
Then starting from the 2 on the right edge we have three forced dominos due to cells with only one available neighbour:
+---+---+---+---+
| |
| 0 0 1 3 |
+ +
| |
| 1 1 0 3 |
+---+ +---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Step 3
If there's a 3-3 at the top-right then we force the other 3-3 to split up:
+---+---+---+---+
| | |
| 0 0 1 | 3 |
+ +---+---+ +
| | | | |
| 1 | 1 | 0 | 3 |
+---+ + +---+
| | | | |
| 2 | 3 | 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
and we end up with two 0-1s in the top-left. Therefore we split the 3-3 in the top-right:
+---+---+---+---+
| | |
| 0 0 | 1 3 |
+ +---+---+
| | |
| 1 1 | 0 3 |
+---+ +---+---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Step 4
This forces the other 3-3 and the top-left square can only form distinct dominos in one way:
+---+---+---+---+
| | |
| 0 0 | 1 3 |
+---+---+---+---+
| | |
| 1 1 | 0 3 |
+---+---+---+---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Alternatively,
Step 3 alternate
If we split the 0-0 in the top-left then the leftmost one forms a 0-1 and the rightmost one would also form a 0-1 whichever remaining neighbour it takes:
+---+---+---+---+
| | |
| 0 | 0 1 3 |
+ + +
| | |
| 1 | 1 0 3 |
+---+ +---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Therefore the 0-0 is forced:
+---+---+---+---+
| | |
| 0 0 | 1 3 |
+---+---+ +
| |
| 1 1 0 3 |
+---+ +---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Step 4 alternate
This forces a chain of two more dominos through cells with only one available neighbour:
+---+---+---+---+
| | |
| 0 0 | 1 3 |
+---+---+ +
| | |
| 1 1 | 0 3 |
+---+---+---+---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Step 5 alternate
Since we now have a 3-3 we must split the 3-3 in the top-right, completing the solution:
+---+---+---+---+
| | |
| 0 0 | 1 3 |
+---+---+---+---+
| | |
| 1 1 | 0 3 |
+---+---+---+---+
| | | |
| 2 | 3 3 | 2 |
+ +---+---+ +
| | | |
| 3 | 2 2 | 1 |
+---+---+---+---+
Here's the image with a coordinate system for easier discussion. (I don't know how to straighten it out while I'm at it.)
observations
Position A1 is very constrained, so start there. We cannot make the A1-B1 domino, because then any domino including A2 would be a duplicate and duplicates are not allowed. Therefore A1-A2 is a domino.
If we make A3-B3, then that would force A4-B4. That in turn would mean we would have to break up C2 and C3, which have many proximate duplicates. Let's pencil in A3-A4 instead. We might need to revisit based on other tiles.
B1-B2 and B3-B4 are unique and have no risk of overlaps, so we'll pencil those in.
We have to be careful of all those 2-pip and 3-pip cells near each other. We did not create A4-B4, so we are free to use C2-C3 (two 3-pip cells). If we do that, we can similarly do D2-D3, leaving C1-D1 and C4-D4.
Checksum: our eight dominos are: 0-0, 1-3, 1-1, 0-3, 3-3, 2-2, 2-1, 2-3.
or, drawn
(Yeah, without being able to straighten out the image I could only hand-draw these lines, so this is not elegant.)
2 comment threads