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# How Many Intersections?

## Question

I have a math puzzle I just came up with. A mesh is formed by 3 sets of parallel lines on a plane (2D surface). Each set contains an infinite number of lines with a constant, perpendicular distance between them, and each line is infinitely long. My puzzle asks that you find an expression for the average number of intersections per area.

Here is a link to my puzzle: http://ericboy.wordpress.com/2012/08/30/mathcritical-thinking-puzzle-83012/. Please let me know what you think or if you have any questions.

Thanks, and Good luck! ## Recommended Posts

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Using the notation in the link:

1/[d1d2csc(θ1)]+1/[d1d3csc(θ2)]+1/[d2d3csc(θ21)]

or

sin(θ1)/(d1d2) + sin(θ2)/(d1d2) + sin(θ2 - θ1)/(d2d3)

As the area of the plane approaches infinity one intersection corresponds to one 'block' or 'section' of the grid made by two intersecting sets of parallel lines. So calculating how many of those 'blocks' from each pair of intersecting sets fit into one unit of area I got the above equation.

Edited by kbrdsk
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Using the notation in the link:

1/[d1d2csc(θ1)]+1/[d1d3csc(θ2)]+1/[d2d3csc(θ21)]

or

sin(θ1)/(d1d2) + sin(θ2)/(d1d2) + sin(θ2 - θ1)/(d2d3)

As the area of the plane approaches infinity one intersection corresponds to one 'block' or 'section' of the grid made by two intersecting sets of parallel lines. So calculating how many of those 'blocks' from each pair of intersecting sets fit into one unit of area I got the above equation.

Yes, you are correct! Well done! Thanks for using the 'spoiler' option to allow others to solve this as well.

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