Math Doodling: Magic of perfect squares

I know! I know! I am perfectly jobless. Somehow these perfect squares never fail to fascinate me. You might be already knowing this and great mahaans would have already discovered this but I take pride in posting this in my blog. Certain facts are really intriguing. If someone could explain, I would be grateful.

Take a look at the chart below:

GROUP 1 : DIFFERENCE 1
4 X 4= 1|6
5 X 5= 2|5  also (1+1)|5
6 X 6= 3|6  also (2+1)|6
7 X 7= 4|9  also (3+1)|9
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GROUP 2: DIFFERENCE 2
8 X 8=   6|4
9 X 9=   8|1 also (6+2)|1
10X10=10|0 also (8+2)|0
11X11=12|1 also (10+2)|1 and so on
12X12=14|4
13X13=16|9
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GROUP 3: DIFFERENCE 3
14 X 14= 19|6
15 X 15= 22|5
16 X 16= 25|6
17 X 17= 28|9
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GROUP 4: DIFFERENCE 4
18 X 18= 32|4
19 X 19= 36|1
20 X 20= 40|0
21 X 21= 44|1
22 X 22= 48|4
23 X 23= 52|9
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GROUP 5: DIFFERENCE 5
24 X 24= 57|6
25 X 25= 62|5
26 X 26= 67|6
27 X 27= 72|9
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GROUP 6: DIFFERENCE 6
28 X 28=   78|4
29 X 29=   84|1
30 X 30=   90|0
31 X 31=   96|1
32 X 32= 102|4
33 X 33= 108|9
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AND SO ON...

Notes:

1. Just  take a second look at the ones marked in red, thats where the shift happens.

2. Whats with the 4 and 6. Cardinality of even numbered groups is 6 and odd numbered groups is 4.

4. Again coming back to the red ones. Take a look at the squares 49, 169, 289, 529, 729. Leave out the 9 we get 4, 16, 28, 52, 72,..... Again divide the series by 4 we get 1, 4, 7, 13, 18,.... Finding out the difference between two consecutive terms, 3, 3, 5, 6, 7, 9,.... See the alternate terms, odd number series and multiples of 3.

5. In the units places of odd numbered groups we have 6, 5, 6, 9...6569 is  a prime no, in even numbered groups 410149 which is also prime. I dont know what to make of this.

6. If you find some more interesting facts based on this please add in comments. I am still searching for more.