Let's say I have x unique numbers and I choose one number y out of this set. Is it possible to create these x numbers such that the absolute difference between y and any other number in $x$ will always be the same? This should work for every value of y in x.
This is trivial for $x = 2$ ...
Numbers: -1, 1
|(-1) - 1| = 2
|1 - (-1)| = 2
But with $x > 2$ this seems more complicated (or impossible?). What are your thoughts?
Is it possible to accomplish this with $x > 2$ so the difference is at least almost the same?
The answer is yes.
Take three numbers $a, b, c.$ Say that $a+b=c$. From that information, we can make a set of fact families:
$$a+b=c$$ $$b+a=c$$ $$c-b=a$$ $$c-a=b$$
These are the traditional equations in a fact family. But we can add one more.
$$b-c=-a$$
This is the most important one for our problem. We know it is true, because we can subtract $c$ and subtract $a$ from both sides of the first equation.
Now we can look at two important equations:
$$c-b=a$$ $$b-c=-a$$
These have a lot of relation to our first problem. We can put absolute value in:
$$|c-b|=a$$ $$|b-c|=a$$
And there you have it. No matter what $b$ and $c$ are, the two equations above will always be true.