Using your Head is Permitted

March 2012 riddle

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 sub-shapes. 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 non-decomposable tiling?

One last point:
This month, more than usually, please make sure your proofs are elegant. It is tempting, in this riddle, to provide a proof along the lines of "I checked all possibilities". When you do something like this, consider:

  1. The list of possibilities needs to be specified explicitly.
  2. For each item on the list you must provide a rigorous proof.
  3. Most importantly: you need to prove that your list of possibilities does not miss a possibility you did not take into account.
In short, please avoid "computer-checkable" proofs. I will accept proofs only when I can verify their correctness.

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 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.


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