You are in charge of giving land to different numbers of individuals and below is how you have divided the land into 10 areas.
You can assign any number of people from 1 to $n$ to live in each area under these conditions, that no neighbouring areas have the same difference in number of people living in them and no area is allowed the same number of people living there as another area.
Neighbouring areas are defined as sharing at least one horizontal or vertical border, diagonally touching doesn't count.
What is the lowest value of $n$ so that the conditions are met? Proof it is lowest.
So from the drawing you can see there are 14 borders which means the minimum value for n=15 so you can have differenced 1,2,3...14. This is as far as I have got. I can allocate numbers to the rest so that it works but I don't know how to go about proving it is the lowest. I haven't managed to get to n=15 otherwise I could have proven it.