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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?

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.

More Total Totality is a follow-up short problem to this one.

*This problem is taken from the UKMT Mathematical Challenges.*