Problem #414

Convex numbers
Public 09/05/17 8xp Programming 24.0%

Take the number 134431001 as an example. We can make pairs using the 2 digits following one another:
[(1, 3), (3, 4), (4, 4), (4, 3), (3, 1), (1, 0), (0, 0), (0, 1)]

If we assume that these are the lattice points, we can draw line segments in the same order and we can have a polygon.

In general for a k-digit number like we can have a (k-1)-gon:
[(d1,d2), (d2,d3), ..., (dk-1,dk)]

If the drawn polygon is a convex one then we will call this kind of number as convex number.

How many positive convex numbers are there?

Answer format: count,sum

[My timing: < 1m]

- Any two neighbouring line segments cannot be collinear.
- All the generated pairs must be distinct.
- Some polygons can be generated by more than one number. 
  For example: 112321, 123211, 232112, 321123, 211232
  Since the numbers are different, they are all counted.

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