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Suppose you draw a collection of lines. What is the maximum finite number of intersections possible?
Hint: To ensure the maximum number of intersections no lines can be parallel and no more than two lines can intersect at a single point.
One line
Zero intersections
Two lines
One intersection
Three lines
Three Intersections
Four lines
Six intersections
Solution Example 1:
Solution Example 1:
Do you see how the number of intersections increases by consecutive integers? We've seen this pattern before! In fact, once we get past the first line and we have points of intersection, just a slight rearrangement and those points can form the triangular numbers! Below I've illustrated the sequence using a "blank" point to represent zero intersections in the first figure.
The maximum number of finite intersections has the same solution as the Triangular Numbers and the Sum of Integers problems! It is just shifted so that it starts at zero instead of 1. If we let n start at 0 instead of one, we can use the same formula as those two solutions. How would we need to adjust the formula if we still wanted to start the first term of the sequence at 1?
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