Tricopter vs Quadcopter vs Hexacopter

When thinking through the three different designs, I am pondering what would be the desirable traits of a Tri, Quad, or Hex, if the overall flight capability of each were the same in terms of the same lift, endurance, and speed.

Now I feel I should ask if these three configurations actually could be made to lift the same for the same amount of time at the same speed.

What makes any designer choose a tri, quad or hex? Does each design have distinct attributes that the others lack? Are they each suited to a specific type of task? Has this been explored in the past in a published article?

Thanks for input from all who have found some answers or studied this for themselves.

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