Who is the original author of this simple paradoxical decomposition? Paradoxical decompositions of sets usually require the axiom of choice; Hausdorff or Banach-Tarski are well-known examples. A paradoxical decomposition of a point set without the axiom of choice has been constructed by Sierpinski and Mazurkiewicz. A set $S$ is the union of two sets $A$ and $B$. When the elements of $A$ are rotated ($\rho$) by one radian, then $\rho$$A = S$, and when the elements of $B$ are translated ($\tau$) by one unit, then $\tau$$B = S$ too.
There is a simple variant. Decompose the set $\mathbb{Z}$ of all integers into $A$, the set of even integers, and $B$, the set of odd integers. When the elements of $A$ are divided $(\delta)$ by 2, then $\delta$$A = \mathbb{Z}$. When the elements of $B$ are translated by one unit (in positive or negative direction) and then divided by 2, then $\delta\tau$$B = \mathbb{Z}$.
Same can be shown for other sets $S$, for instance the set of positive integers (then $B$ must be translated by +1).
My question: Have these paradoxical decompositions already appeared in literature? I would like to include them into my lectures with appropriate quotation but could not yet find a source or an author.
 A: This is not a full answer, but the reference may be interesting nonetheless. Paradoxical decompositions of sets are discussed in the very beginning of Stan Wagon's book "The Banach-Tarski paradox", Cambridge University Press, 1985 (very recommendable, lots of paradoxes, nice for lectures). The following is a nice quote from the first paragraph of the first chapter:

In a famous example, Galileo observed that the set of positive integers can be put into a one-one correspondence with the set of square integers, even though the set of non-squares, and hence the set of all integers, seems more numerous than the squares.

Ok, this is slightly different from the Hilbert hotel situation, but the basic idea seems to appear as early as 1638. 
The more general statement (Wagon calls it "the modern version of Galileo's observation") appears as Theorem 1.4 of Wagon's book: a set $X$ is paradoxical with respect to the action of its permutation group if and only if it is infinite. 
Unfortunately, the exact history of this result is not discussed in Wagon's book (the name Tarski appears, though). The if direction requires the axiom of choice (but not for cardinals).
