Recognizing patterns!
Probability
Level
2
How many diagonals does a regular -gon have?
3141592
2481387.
2033135.
2036162.
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Consider the 2018 vertices without any of the edges or diagonals connecting them. Now, let's draw all edges and diagonals: The first vertex gets connected to 2017 other vertices. The second gets connected to 2016 (it's already connected to the first), the third to 2015, and so on. 2 0 1 7 + 2 0 1 6 + ⋯ + 1 = 2 2 0 1 7 × 2 0 1 8 = 2 0 3 5 1 5 3 Now, 2018 of the lines we have drawn are edges of the polygon, and the rest are diagonals. Thus the number of diagonals is 2 0 3 5 1 5 3 − 2 0 1 8 = 2 0 3 3 1 3 5