Let us consider two integer rectangles (that is, with sides of integer length) $S$ and $T$ of the same area. Then, obviously, $S$ can be dissected into several integer rectangles $A_1$, ..., $A_n$ (we will call such a dissection a tiling, and the rectangles constituting it tiles) which can be rearranged to form $T$ (e.g., dissect $S$ into unit squares).
Question #1. Is there any non-constant lower bound (expressed as some function of dimensions of rectangles $S$ and $T$) for the size of such a tiling?
(Size of the tiling is its cardinality, the number of the tiles it consists of)
This sounds rather vague. Also there is an issue of the dimensions of $S$ and $T$ having common divisors. Let me formulate a very specific question (perhaps even somewhat too specific but...).
Question #1.1. Let $p < q < r < s$ be four different prime numbers, $S$ --- rectangle with dimensions $pq \times rs$, T --- rectangle with dimensions $pr \times qs$. Does there exist a function $f(p,q,r,s)$ which a) serves as a lower bound for the above-mentioned tiling size; b) tends to infinity if all four arguments tend to infinity?
A possible answer: $f(p,q,r,s) = h(p)$, where $h:\mathbb N \rightarrow \mathbb N$ tends to infinity as its argument tends to infinity (like $\log p$, for example).
Another (similar but diferent) interesting question of a similar nature (also quite specific):
Question #2. Given a natural number $n$, let $S$ be the rectangle with dimensions $n \times (n+1)$, and $T$ the rectangle with dimensions $(n+1) \times n$. Is there a non-constant lower bound (as a function of $n$) for the size of a tiling of $S$ such that its tiles can be moved without rotating them to form $T$?
Perhaps we should concentrate first on the following simplified version of Question 2.
Question #2.1. If $n = 1000000$, then is it possible to dissect an $n \times (n+1)$ rectangle into less than $20$ integer rectangles which can be moved (without rotation) to form an $(n+1)\times n$ rectangle?
Note. Don't try to dissect into square tiles. It can be proved (although the only elementary proof I know is neither very short nor very elementary) that $(n+1) \times n$ rectangle cannot be dissected into less than $\log_2(n)$ square tiles.