The following riddle originated from an open question sent to me by
R. Nandakumar. (Thanks!) The full history of the evolution of this question and
how it was solved will be given on the solution page. (Some other variations on
the question will also be presented there.)
This month's riddle is composed of two parts. Answer either or both for credit, and provide proofs. A separate solvers' list will be kept for each part. Both parts this month deal with the problem of tiling a rectangle into N rectangles, with N>1. The rectangle sides need not be of rational lengths. (A tiling is a partitioning of the original shape area into subshapes. Overlaps and unallocated areas are not permitted.) Part 1:For which N values is there a tiling into rectangles of equal area but different perimeters?Note: you can choose the dimensions of the rectangle you tile, including a different rectangle for each N. Part 2:Let a tiling be decomposable if there is some subset of M rectangles, 1<M<N, that forms, on its own, a tiling of a rectangle (inside the tiling provided by all N rectangles).For which N values is there a nondecomposable tiling?
One last point:

List of solvers:Part 1:Adam Daire (13 March 19:37)Ganesh Lakshminarayana (27 March 04:56) Part 2:Lu Wang (7 March 15:48)Adam Daire (8 March 00:46) 
Elegant and original solutions can be submitted to the puzzlemaster at riddlesbrand.scso.com. Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.
The solution will be published at the end of the month.
Enjoy!