Our design team drew all 48 line segment?

Question

Our design team drew all 48 line segment that connect 6 points on one side of rectangle R to 8 points on the opposite side. In at most how many different points inside R can those 48 line segments intersect?

Answer 1

I drew and counted up to 5x5:

× | 2 | 3 | 4 | 5

-- | -- | --| -- | --

2 | 1 | 3 | 6 | 10

3 | 3 | 9 | 18 | 30

4 | 6 | 18 | 36 | 60

5 | 10 | 30 | 60 | 100

Figuring that 2x would continue with triangular numbers (15, 21, 28 ...)

and that the same ratios would hold

Row 3 = 3 x Row 2

Row 4 = 2 x Row 3

Row 5 = 5/3 x Row 4

Row 6 = 3/2 x Row 5

= 15 45 90 150 225

Row 7 = 7/5 x Row 6

= 21 63 126 210 315

Row 8 = 4/3 x Row 7

= 28 84 168 280 and .... your answer 420

This assumes that all intersections are distinct.

By the time you reach that number it gets pretty crowded!

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