(In reply to
Puzzle Answer by K Sengupta)
In general, we consider the square of the n-digit number 666....668 as follows:
66......668^2 = {(2*10^n +4)/3}^2 = (1/9)*{4*(10^2n) +16*(10^n)+16}
n-1 6s
= (1/9)*{4*(10^2n)-1)) +16*(10^n -1)+36}
=4*(10^2n -1)/9) + 16*(10^n -1)/9 +4
= 44......44 +16*(11.....11) + 4
2n 4s n 1s
= 44.....44 + 177......776 +4
2n 4s n-1 7s
= 44.....44622.....224
n-1 4s n-1 2s
Then, the sum of the digits is:
4(n-1) + 6 + 2(n-1) + 4
= 6n+4
In the present problem, we have:
n-1=10^5-1
=> n = 10^5
Consequently, the required sum of digits
= 6*(10^5) +4
= 600,004
Q E D
Edited on May 28, 2022, 1:10 am
Explanation to Puzzle Answer
