Imagine you are given a chain made of or more pearls by a king and asked to find all the unique shapes of the chain without breaking it. The order of pearls must remain the same, and therefore you can only change the connection of each adjacent pair of pearls, which can be either straight or perpendicular. Rotations and reflections are considered as the same shapes looked at from different angles.
The unique forms of chains with pearls are shown in the table.
What is the number of unique forms for pearls?
Note: It is not as easy as it seems to get all possible shapes. Therefore, please think hard.
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Here is how to do it the easy way;
Parallel to A---B (as A---B---C)
Above B in A---B
Below B in A---B (this form would be simply a reflection of above second form therefore we neglect it)
Therefore there are only two unique forms of the chain formed with 3 pearls.
Following are all the possible forms or shapes when we have five pearls;
As we see here, there are 1 3 unique structures or forms of the chain formed with 5 pearls.