1
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Absolutely No calculators or computers.

Only Simplest Method for finding the missing digits will be accepted as the right answer.

Given:

The square of this 21 digit number

$ 229 359 782 235 085 482 225 $

is a 41 digit

Palindromic Square:

To emphasize the palindromic nature of the square, it is split as 2 lines. Top line shows first 20 digits and the swing digit 2 as the 21st. Second line gives the last 20 digits in reverse order.

$ 5xx xxx xxx 069 258 339 64 $ $2 $

$ 5xx xxx xxx 069 258 339 64 $

To avoid confusion, I am giving the full number from left to right also:

$5xx xxx xxx 069 258 339 64 $ $2$ $46 933 852 960 xxx xxx xx5$

Find the missing digits xxxxx...x can be any number from 0 to 9.

If you know the right method ...5 minutes or less.

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The answer is (absolutely no counting devices, about 8 mins for counting + about 2x time for writing and formatting the answer :) )

26059097, i.e. $$229359782235085482225^2 = 52605909706925833964246933852960790950625$$

Explanation

The number ends with 25, so it should be of the form $200x + 25$, because there is an even digit before 25. We get $(200x+25)^2 = 40000x^2+10000x+625 = 10000(4x^2+x)+625$. It immediately means that the number ends with 0625 (so we get 260). To calculate the other 5 digits, note that $2x$ ends with 54822, so $x$ ends with 27411, and $4x^2=(2x)^2$ ends with 51684. The latter result as achieved by doing a partial multiplication keeping only the last 5 digits (it's easy to do):
54822
54822
--------
09644
9644
576
88
0
--------
51684
So, the other missing digits are 59097, reverse of $51684+27411=79095$.
P.S. We can find the (full) actual value of $x=1146798911175427411$ very quickly, but this is not needed for the solution.
P.P.S. The process can be optimised by using the $2000x+225$ representation in the 1st step to get 50625 at the end, and then multiplying 4-digit numbers.

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  • $\begingroup$ Good.but even simpler method is there. $\endgroup$ – Uvc May 17 at 9:39

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