FAQ Answers
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Problem 2
Is the figure made of pentabends allowed to have a hole?
If you can arrange the pentabends into some sort of loop shape with some empty space in the middle, then there would be two disconnected components of the boundary, and their lengths would be added up when computing the roundness.
Can we form different pentabends by placing the arcs at different angles to each other?
No. The only pentabend is precisely the one drawn in the picture (or rotations, translations, and reflections of it). One way to specify this figure is to say that:
- The top three vertices are all on the same line; call that line L.
- The five vertices are symmetric about the line through the middle vertex and perpendicular to L.
Problem 4
In the definition of arithmetic sequence, are we allowed to pick the values of the integer n?
No. To unpack the set-builder notation in the problem a bit, "{an + b : n ∈ ℤ} with a, b ∈ ℤ and a ≠ 0” means that to form an arithmetic sequence, you first pick some a and b as described, and then take the set consisting of an + b as n varies over all integers. So, for instance, {0, 2, 4} is not an arithmetic sequence. We could attempt to build it by setting a=2 and b=0, but then this set would only consist of the values at n=0,1,2; the full arithmetic sequence looks like {..., -4, -2, 0, 2, 4, ...}.
