A perimeter of the square is 40 cm. Draw as many rectangles as you can into that square. All rectangles must have different perimeters, and integer side lengths.
(So far, I got 12)
Here is a hint on how you may check your answer:
If the perimeter of your square is 40, then each side is 10.
This means you can draw a 10 by 10 square and imagine 100 small equal squares inside. Now try to find all the possible unique rectangle shapes.
(check your answer again there should definitely be more than twelve.
I think the different perimiters part is the key to solving this problem efficiently...
The maximum perimiter that any rectangle can have is 40, and you can only draw one of those.
The next biggest perimiter would be 39, but you can't draw a rectangle with integer side lengths with that perimiter.
The next one would be 38, you can draw lots of rectangles with that peremiter, but only one will count because they have to have different perimiters.
37 is out because its odd
36 is in because its even
all the way down to a peremiter of 4, which is the smallest rectangle that meets the restrictions
There are 40/2  1 = 19 possible rectangles.
Goodbye, square! Hello, rectangle!
a and b do not have to be different in order for it to be a rectangle. A square is a rectangle.
Never heard of it. When I went to school, some 55 years ago, a square was a square, and a rectangle was a rectangle.
One mathematical definition of a rectangle is "a quadrilateral with four right angles."
From: https://en.wikipedia.org/wiki/Rectangle (You can find other sources with effectively the same definition.)
A square meets the conditions needed to be considered a rectangle. A square is just a special case of a rectangle.
Math has improved and developed over time, guest. There are new concepts being taught in school. As our civilization and technology improves, our school must keep up with it.
Euclid gave us a definition of a "rectangular parallelogram" – a "rightangled parallelogram" which we now call a "rectangle" – that does not exclude squares, and this was written around 300 BC (much longer than 55 years ago!).
See:_  Elements Book II  _and_  Elements Book II Definitions 1 and 2  
And:  Elements Book I  and  Elements Book I Proposition 34  (where he talks about a parallelogram) 
 

LOL Hectictar is right.
A square has every property that a rectandle must has (plus more) therefore it is a rectangle.
It is like this:
A terrier is a dog. It has every requirement to be a dog (but it has extra requirements as well.)
Therefore, although all terriers are dogs, not all dogs are terriers.
All squares are rectangles but not all rectangles are squares.