Mathematically prove a concept

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Hi, I've come up with a concept for a mini-game, which requires the user to connect objects with lines, and each object needs to have a line to it with 2 intersections (on the line). I wish to determine what the optimal solutions for any kind of configuration of objects is through some sort of proof.

In a 2D plane, place stars, squares and circles, then draw lines between them. The stars and squares are fixed, while the amount of circles you add should be minized. All stars must have a line to it that is crossed twice. As many squares as possible need to have a line to them that is crossed twice. From a square to the star, a single line can be drawn which should be crossed twice (circle -> square -> star - for bonus). Circles can have up to two lines coming from them, and squares have up to 1 line coming from them.

I need to know what locations the circles should be drawn and where they should be connected to with lines to achieve the above.

Matematiikka Algoritmi

Projektin tunnus: #28432303

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