bit of an interesting problem I have here!
Here it is:
Is it possible to choose ten different numbers from the set {0, 1, 2, . . . , 14} and place them in the ten circles in the figure below in a way that all positive differences between the pairs of numbers in adjacent circles (i.e., circles connected by an edge) are all to be different.
Alright, so when approaching this problem, the first thing I note is that there are a total of 14 edges. The maximum difference between 2 points is 14 and the minimum difference is 1. That means there are 14 possibilities for differences between 2 points. That means that if this is possible, for every number 1, 2, 3, ... 14 there will be exactly one edge with that number as its difference.
We can also conclude that 14 and 0 must be connected.
Similarly either 13 and 0 are connected or 14 and 1 are connected.
12 and 0 are connected, or 13 and 1 are connected or 14 and 2 are connected.
You can continue this pattern to see all the possible points that will produce each difference.
Now I am a little bit stuck here. Something tells me that this is not possible (I have not been able to create a successful solution), although I can't quite be certain of why. I see a lot of pieces of the puzzle here, but I can't seem to fit them together.
Any help is much appreciated! Thanks in advance.
If you look carefully at the diagram, every circle is adjacent to two or four other circles. Concentrate on one circle, say it contains the number $a$ and is adjacent to two other circles containing numbers $x$ and $y$. Then the differences on the two edges are $$a-x\ \ \hbox{or}\ \ x-a\qquad\hbox{and}\qquad a-y\ \ \hbox{or}\ \ y-a\ .$$ If you add up the two differences, $a$ is counted twice ($a-x$ and $a-y$), or twice negatively ($x-a$ and $y-a$), or not at all ($a-x$ and $y-a$, or $x-a$ and $a-y$). So altogether, $a$ is counted an even number of times. The same holds true if there are four adjacent circles; and the same holds for all numbers besides $a$. So, if you add up all the differences, you must get your ten numbers an even number of times each, for an even total. So the differences cannot be $1,2,\ldots,14$, as these have an odd total.