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.

Kaprekar numbers- 7 digits

There is no single no of 7 digits but a series of 8 numbers as follows-
8429652...7619733..8439552...7509843..9529641..8719722..8649432...7519743...8429652(rpt)
Few of them explained below-
a)7619733. Highest number 9776331...Lowest number.1336779..Difference 8439552
b)8719722 .Highest number 9877221..Lowest number  1227789..Difference 8649432
c)7519743  Highest number 9775431..Lowest number  1345779..Difference 8429652

Kaprekar numbers-6digits

In my previous post of 26 oct 2012,I had shown 2 series of Kaprekar numbers of 5 digits as follows-
74943,62964,71973,83952,74943(repeat)
61974,82962,75933,63954,61974(repeat)
To enable you to recollect,the process is explained below-
74943.-This consists of digits 3,4,4,7,9
Find the largest and smallest numbers with these and find the difference.You will get the next number in the series viz 62964 and so on.
viz 97443-34479=62964
Now I will show two 6 digit numbers and one series of 6 digit numbers-
1)Number  631764-the process results in the same number..766431-134667=631764
2)Number 549945-behaves like 631764
3) Series-851742,750843,840852,860832,862632,642654,420876,851742(repeat)