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UPDATE: I misunderstood! The rods have to be in the order 1 ... n, so the solutions in my picture aren't really solutions! This makes the puzzle a lot harder.

A fun question to start the day from Ali Sada on the math-fun mailing list:

"What are the triangular numbers, ( T(n) ), such that a chain of rods with lengths 1, 2, ..., ( n ) (connected by hinges) can be arranged to form a rectangle with a perimeter exactly equal to ( T(n) (i.e. using each rod exactly once without any loss or overlap)?"

If I've understood it correctly, here are two solutions to get started: