# Fill the circles with the numbers 4, 5, 6, ... 10

Using all the numbers 4, 5, 6, ... 10 exactly once, fill each empty circle with a number so that if two numbers are in circles that are joined by a line then their difference (in absolute value) is at least 3.

• Is my solution missing anything, or leaving out anything? Jul 18 at 19:24
The top rectangle must have $$4$$, $$7$$ and $$10$$, as all the numbers are connected to each other and have to be 3 apart. But $$7$$ has to go in the top circle because of the requirement in the bottom: The node under $$7$$ would have to be either $$4$$ or $$10$$, and that would be impossible.
So then we can deduce the rest of the graph: If $$10$$ went in the left node, then the node under it would have to be either $$6$$ or $$5$$. Therefore the bottom-left corner node has to be $$9$$ or $$8$$. Similarly for the right node, $$4$$, the node under that would have to be $$9$$, since if it were $$8$$, the node under that would not have any number to go with. So then the node under that has to be $$6$$.
Now, the bottom-left corner node must be 8 since 9 is used up. But that is impossible since that violates the requirement. Henceforth, contradiction, meaning $$4$$ has to go in the left node and $$10$$ has to go in the right node.