Finding square numbers

Everyone knows how to extract the square root of any number
My proposal here is to find a number which has certain specialities but is also a perfect square number
For instance you are asked to find a number starting with six 2's and which is a perfect square.
With a little work of trial and error you can find it.
My research gave the number 222222674025 which is the square of 471405
But suppose I ask you to find a square number starting with five 2's.Try to do a little math and find the number.
Will it be smaller than 222222674025 shown by me earlier?

Collecting numbers for particular sums

Given below are 2 groups of numbers-
Group A-17,32,55,41,26,67,11,37
Group B-27,45,31,38,49,7
Collect 6 numbers from Group A for a sum S1 and 4 numbers from Group B for a sum S2,where S1/S2=2
You can find 3 solutions at least.Proceed

Prime pyramid-Adjacent primes

You are to work out a number pyramid starting with 1 number in the first row,2 in the second,3 in the third,4 in the fourth row and so on.Each row should start with digit 1 and end with the digit represented by the row,for example the 7th row should end with 7.The digits/numbers to be filled up in between should be such that each pair of adjacent numbers should sum up to a prime.A few rows are shown filled up below-
1-                       Sums involved  -nil
1,2                                             3
1,2,3                                          3,5
1,2,3,4                                       3,5,7
1,4,3,2,5                                    5,7,5,7
1,4,3,2,5,6                                 5,7,5,7,11
1,4,3,2,5,6,7                              5,7,5,7,11,13
1,6,7,4,3,2,5,8                           7,13,11,7,5,7,13
Try filling up subsequent rows.There could sometimes be more ways in respect the digits/numbers involved in the middle of each row.
It may not be easy when you proceed upto row 17,19 etc.

Surprise in sudoku-rejoinder

Were you able to fill up the sudoku grid shown by me?
The completed grid will be as shown below-(row-wise)
418 239 657
695 781 342
273 546 819
961 853 274
354 627 981
782 914 563
546 192 738
839 475 126
127 368 495
I have deliberately shown the digits in groups of three
There is a  surprise in rows 1,3,5,7 and 9 where you will notice an addition sum as follows-
418+239=657
273+546=819
354+627=981
546+192=738
127+368=495 

Probability for division of one number by another

Choose any number (A)between 10 and 1000.Remove the last digit in the same to get another number (B).For example if A is 456 ,B will be 45.What is the probability that A will be divisible by B?

A sudoku with a surprise

Here is the grid for filling up by you-should not be difficult-The cells which were left blank have been marked b-
bbbbb96bb
6b5b8bbb2
bbbb4bbb9
b6185bb7b
b5bbbbb8b
b8bb1456b
5bbb9bbbb
8bbb7b1b6
bb73bbbbb
Fill it up and find out what the surprise is. I cannot reveal it now as you will be able to fill up the grid more easily.

Sums of 2 squares reading to totals 'c' and '2c'

If a number 'c' is a sum of 2 squares of say 'x' and 'y',can '2c' be also expressed as a sum of 2 different squares?
The answer is yes.
You have to work out (x+y) and (x-y) ,find their squares and add up.
We already have c=x^2+y^2
We now have (x+y)^2+(x-y)^2= (x^2+2xy+y^2)+(x^2-2xy+y^2)=2(x^2+y^2)=2c
Example-Let x=17 and y=9.So c=289+81=370
We also have (x+y)=26 and (x-y)=8.Their squares are 676 and 64 giving a total of 740 which is the same as '2c'.
As a corollary if we have three numbers x,y and z and their squares add upto 'c',we can express '4c' as the sum of 6 squares viz of x+y..x-y...y+z...y-z....z+x and z-x.