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## 'Doodles' printed from http://nrich.maths.org/

A "doodle" is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be "double points"
not "triple points". Number the vertex points in any order.
Starting at any point on the doodle, trace it until you get back to
where you started. Write down the numbers of the vertices as you
pass through them. So you have a [not necessarily unique] list of
numbers for each doodle.

Prove that

- each vertex number in a list occurs twice. [ easy ! ! ! ]
- between each pair of vertex numbers in a list there are an even
number of other numbers [ hard ! ! ! ]