Prime pyramid-adjacent primes-2

This is with reference to my earlier blog-same title
I had shown row 8 as follows-1,6,7,4,3,2,5,8 where each pair of adjacent numbers add upto a prime viz 1+6=7,6+7=13,7+4=11,4+3=7,3+2=5,2+5=7,5+8=13
The further rows can be formed as follows-
a)As 8+9=17 (a prime) ,we can simply add 9 to get 1,6,7,4,3,2,5,8,9
b)As 9+10=19(a prime),we can simply add 10 to get 1,6,7,4,3,2,5,8 ,9,10
c)As 10+11=21 which is not a prime we have to modify numbers in row (b)
For this we follow the same position upto 3,add the nos from 2 to 10 in the reverse way and then add 11
We will get 1,6,7,4,3,(10,9,8,5,2),11
d)12 can be added as 11+12=23,giving us 1,6,7,4,3,10,9,8,5,2,11,12
e)13 cannot be added as 12+13=25 is not a prime.The change is made as follows-
1,6,7,(12,11,2,5,8,9,10,3,4),13
Further rows are filled up in the same fashion-
f) 1,6,7,12,11,2,5,8,9,10,(13,4,3),14
h)1,6,7,12,11,2,5,8,9,10,13,4,3,14,15
j)1,6,7,12,11,2,5,8,9,10,13,4,3,14,15,16
k)1,6,7,12,11,2,5,8,9,10,13,4,3,(16,15,14),17
l)1,6,7,12,11,2,(17,14,15,16,3,4,13,10,9,8,5),18
m)1,6,7,12,11,2,17,14,15,16,3,4,13,10,9,8,5,18,19
n) 1,6,7,12,11,2,17,14,15,16,3,4,13,10,(19,18,5,8,9),20
We can follow the same procedure for further rows.  

Division one number by another-probability-rejoinder

 This is with reference to my previous blog where we consider any number A between 10 and 1000-remove the last digit to get number B-We wanted the probability of B evenly dividing A.
Total number of choices possible for A----1000-9=991
Following give correct divisibility by B-
1)All numbers ending with zero (like 450)---------109
2)All multiples of 11 below 100 (like 22,33 etc)-----8
3) Six other cases-24,26,28,36,39,48---------------6
Total----------------------------------------------123
The probability is hence 123/991

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?