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 kdigit number like d_{1}d_{2}...d_{k} we can have a (k1)gon: [(d_{1},d_{2}), (d_{2},d_{3}), ..., (d_{k1},d_{k})] 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] Notes:  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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