Series 1:
I would start with 41..Now if we have to form a 4 digit square with 41 there is no suitable pair as 4096 and 4225 are consecutive squares and there is no perfect square starting with 41. So technically the series ends with 41!
41
Series 2:
Now let's choose 37 next
a) 37 along with 21 gives us a perfect square 3721
b) With 21 as the starting number, we could combine it with 16 to get 2116 which is a perfect square - 37 21 16
c) With 16, there are 2 options, 00 and 81 (1600 and 1681 are perfect squares)
37 21 16 00 (Series would come to a stop since technically it becomes a 2 digit square again)
37 21 16 81
d) With 81, we could combine with 00 to get 8100 and the series stops there
37 21 16 81 00
Series 3:
We would go with 70 next. Following the same approach as above, we will get
70 56 25 00
Series 4:
Lets begin with 27 and follow the same logic
27 04 84 64 00
27 04 41
27 04 00
Lets look at two more examples where we start with smaller number like 03 as the starting digits
Series 5:
Beginning number - say 03
03 24 01 21 16 81
03 24 01 21 16 00
03 24 01 44 89
03 24 01 69
03 24 01 96 04 00
03 24 01 96 04 41
03 24 01 96 04 84 64 00
03 61
If we observe, the smaller the starting number, the more combinations we get to see till we reach the end
Drawing a simple analogy to the corporate world...
A bottoms-up approach, where we get all the team members right from the junior most person in the organization, to actively contribute towards the broader vision, would bring in more number of ideas and foster innovation rather than a top down mandate
