Determine number of squares in progressively decreasing size that can be carved out of a rectangle How many squares in progressively decreasing size can be created from a rectangle of dimension $a\;X\;b$ 
For example, consider a rectangle of dimension $3\;X\;8$

As you can see, the biggest square that you can carve out of it is of dimension $3\;X\;3$ and they are ABFE and EFGH
The next biggest square is of dimension $2\;X\;2$ which is GJIC
Followed by two other squares of dimension $2\;X\;2$ which are JHLK and KLDI
So the answer is 5.
Is there any mathematical approach of solving it for a rectangle of arbitrary dimension?

Mathematically, start with the square pyramidal number under the rectangle's area, and compare with the list of known tightly packed rectangles, which usually have low excess values.  You could see if a given rectangle is at squaring.net for known perfect packings.
It's quickly a hard problem.  For example, can a $189\times189$ or $190\times188$ rectangle be cut into 47 decreasing squares?  This problem is too combinatorially hard to solve at the moment.
