Given the pattern above, which shape is next, and why?
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2 Answers
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The next shape should be:
i.e. A nonagon with 8 of its 9 vertices erased.
Why?
The 5 given images are composed of 1, 1, 2, 3 and 5 line segments, which is the beginning of the Fibonacci sequence.
The next shape should therefore be composed of 8 segments, since 8 is the next Fibonacci number.
Moreover, following the pattern of the triangle, square and hexagon, the shape should have 1 more side than the required Fibonacci number. (NB It is not possible to draw the first two terms in this way, them both being equal to 1; hence these are just depicted as lines.) Since this pattern has been achieved by removing all but 1 vertex in these shapes, we follow suit to draw a nonagon (9 sides) with 8 vertices deleted.
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2$\begingroup$ rot13(Vs vg'f ernyyl whfg zbqvsvrq Svobanppv V'q or qvfnccbvagrq ohg vg frrzf cynhfvoyr ) $\endgroup$– AviCommented Sep 7, 2021 at 22:23
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There could be another solution (without taking anything away from the first one):
The number of edges are 1, 1, 3, 4 and 6. It could be seen as a semi-Fibonacci sequence where the next term is $F_{n+1} = F_{n} + F_{n-1} + 3 - n$ instead. As such, the next term would be 8.
As for the connected edges, the first two "shapes" have no such edges, so clockwise the 0th, 0th, 1st, 2nd and 2nd vertices are black in it. The next shape's black vertex is the 3rd one. The shapes whose number of vertices are powers of 2 aren't "tilted" at all, meaning our octagon's top edge would be completely horizontally oriented:
...81...
7.....2
6.....3
...54...
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1$\begingroup$ Or the sequence could be 1, 2, 3, 4, 6, which is obviously a list of the factors of 12, so the next one is obviously 12... $\endgroup$ Commented Sep 8, 2021 at 10:26