Overlapping Binary Relational Sets
Thanks to ShriekingViolet for pointing out this technique and Zenobia43 for spelling it out in detail.
Puzzle Baron's YouTube video on solving techniques has an example of overlapping binary relational sets. THIS LINK will take you to the exact point in the video where it is pointed out (be sure to hit pause so you can read the text).
The video points out that there is a value relationship between two values in the same category, both of which have two potential values, and that these values overlap. In the video this relationship can be written as "$5.35 is either Diet Coke or Orange Soda". This eliminates all other values of sodas for "$5.35".
What ShriekingViolet pointed out IN THIS THREAD, is a technique for making this same overlapping binary relationship between values in two DIFFERENT categories.
I've spelled it out in this graphic:
It essentially boils down to this:
When there are only two potential values for each element in a fixed relational clue, and one value, A, exists for each element, B and C, you can write the relationship as “A is either B or C”.
It's a clue that is as valid and useful as the clues you are given, but it's hidden in the puzzle.
Here's another example. In this one, the overlapping sets are a bit more spread out, but they still overlap.
Note how Kristen and Fredrick are related in Clue 3. Their potential dollar amount values overlap at $4.70. Thus we can say:
$4.70 is either Kristen or Frederick.
Thus, with the exclusionary "X" at the intersection of Kristen and Fredrick, our clue tells us that $4.70 is not Henry or Minnie.
Here's the blank puzzle, because someone is going to want to do this themself:
Chaining clues together to get an Overlapping Binary Relational Set
Here's a more advanced type of Overlapping Binary Relational Sets. Instead of being given a fixed relationship with one clue, we need to chain the clues together to find the overlapping value.
I noticed the chain on my first pass through the clues. Clue 9 renders the effects of this particular example moot. However, I still think this is a very useful tool.
Expandng Overlapping Binary Relational Sets to THREE sets
I found this gem and was able to apply the concept of Overlapping Binary Relational Sets to three sets of data. In this example, three sets of data overlap so that, no matter what the outcome, one value must exists in one of the three sets. Thus we can remove that value from consideration for the remaining values in the category.
Here's the graphic:
This next approach does not work!!!
With the previous method of showing three clues overlapping in one category, it occurred to me to look for puzzles that have three clues overlapping in three categories. I found one. The approach I took DOES NOT WORK.
So why mention it? Because it occurred to me, and it might occur to someone else.
It takes the form of:
Time is either Name or Food or City
Name, Food, and City all overlap so that in the final solution, one of them will be Time. But how to break down the clue into something useful? You'll see my approach in the graphic here:
The puzzle is easily solved using other, more common, techniques. I wasn't stuck at this point, I was merely paused to look at the relationship of the three categories.
If someone else comes up with a solving technique so that "Time is either Name or Food or City", I'd love to hear it
You would need to look at the food and city together. In this puzzle, 12:30 is either Ollie or (the set of people that are possible for *either* oyster *or* Naperville). Look down the left-hand side of the grid at the rows for Naperville and oyster simultaneously. If you could find someone who was Xed out on both oyster/Naperville, you could go ahead and X that person out on the 12:30 line, too (except for Ollie, who in this case is the only one).
I can't remember what the set notation looks like, it's been too long since I had those math classes :) But I'm fairly certain this would work. eta - because it would read as [Person] is neither oyster nor Naperville, and they are obviously not Ollie, so they can't be 12:30.
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