How many differant sized squares whose areas are whole numbers can be constructed on 5 x 5 geopaper?

Question

What are the demensions and the area of each? Look beyond the obvious ones. (4 are dimond shaped)

Answer 1

You would have the obvious ones of 1x1, 2x2, 3x3, 4x4, and, the largest, the 5x5.

Now for the fun of trying to explain how the diamond shaped ones will work. The diamond shape ones will be squares, but with the geopaper in front of you the diamond shaped squares will have one point towards you.

If you look at the asterisks below as the geopaper:

*****
*****
*****
*****
*****

Connect a square using the asterisks that I replaced with x

*****
*****
*X***
X*X**
*X***

I know it's kinda weird to see but that will be a square with side lengths of root 2. The area of that square would be root 2 times root 2 which equals 2. Since this has a whole number area it fits the description of what you are looking for.

Now you need to try to find the other diamond shaped squares by making the diamonds larger.

Hope this helps. Good Luck.

Answer 2

5 different ones, 1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25
The diamond shape is a square. This seems simple to me and I wonder if you've omitted something. Did you mean rectangle instead of square?