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This network has nine edges which meet at six nodes. The numbers $1$, $2$, $3$, $4$, $5$, $6$ are placed at the nodes, with a different number at each node. Is it possible to do this so that the sum of the $2$ numbers at the ends of an edge is different for each edge?
image of 9 edges which meet at 6 nodes
Either show a way of doing this, or prove that it is impossible.

More Total Totality is a follow-up short problem to this one.
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.  

This problem is taken from the UKMT Mathematical Challenges.