Welcome to the staging ground for new communities! Each proposal has a description in the "Descriptions" category and a body of questions and answers in "Incubator Q&A". You can ask questions (and get answers, we hope!) right away, and start new proposals.
Are you here to participate in a specific proposal? Click on the proposal tag (with the dark outline) to see only posts about that proposal and not all of the others that are in progress. Tags are at the bottom of each post.
Comments on Draw the boundaries of the eight dominoes
Parent
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
Post
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