A game of darts (sort of)

Discrete Mathematics Level 5

Sam wants to make a special "game of darts". He has an dart board which is an infinitely big 2D plane. On this plane he throws 7 red darts and \(x\) blue darts. Once he has placed the darts, he draws a red line segment between every single red dart. Similarly, he draws a blue line segment between every single blue dart. Where two blue lines intersect he places a green dart. Similarly, where two red lines intersect, he places a green dart as well. Then, he collects all the green darts he placed and clears the dart board.

Afterwards, he throws the green darts he collected and then he draws a green line segment between every single green dart. Finally, he counts the number of intersections between all the lines and that is his final score. If he wants the maximum score to be less than 1,000,000,000 , what is the maximum amount of blue darts he can have?

Details and assumptions:

  • Sam can throw the darts any way he wants.
  • Line segments are drawn between the darts.
  • \(x\) is a positive integer.
  • Calculators allowed

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