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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

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