Incredible mathematic game
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Dr. C. H. Laws.  sums56@yahoo.com.tw  羅緝熙醫生

This is an incredible mathematic game, that I played with my professor in the SUMS, in the year 1951 - 1952. I was a first year medical student in that time. 

L. =  Dr. C. H. Laws  P. = Professor or Player

L. Professor, please choose a natural number of any size, write it down on the paper and don't let me know.
P. O. K.. 

L. Multiply this number by 21. 
P. O. K.. 

L. " 6 " is the first perfect number, isn't it?
P. Yes, it is. 

L. The factors of 6 are 1, 2, 3 ?
P. Right.

L. Please add the sum of any combination of them to the product above.
    In other word, add 1 or 2 or 3 or 4 or 5 or 6 to the product above.
P. O. K.. 

L. Then multiply the sum by 1 5 8 5 7 1 2 7. 
P. O. K.. This is a number of 16 digits.

L. Please hide five digits of the product marked ( ? ) below and let me know the rest.

-

----?-?--?-?-?-


P. O. K.. Here it is.

10690?3?62?4?0?3

L. O. K.. Let us have a break.   = = = = =
L. Let us continue now.
L. Please choose another natural number of any size and also don't let me know.
P. O. K..

L. Multiply this number by the number by the hidden digits in order. 
P. O. K..

L. Multiply the same number by 1 5 2 6 2 0 9 please? 
P. O. K.. 

L. Add them up please. 
P. O. K..

L. Hide these 5 digits marked (?) and let me know the rest please. 

-??--?---??--

P. Here it is.

9??92?183??33


L. O. K.. I'll tell you all the figures. 
P. Really ?

I found the right answer for my professor in a short time.

A The first chosen natural number was3210344
  Multiplied by 2 1x21
_____________
B The product was67417224
C Added 5+5
_____________
D The sum was 67417229
     Multiplied by 1 5 8 5 7 1 2 7 x15857127
______________________
E The product was 1069043562241083
F 5 digits were hidden 45218
The first incomplete result was 10690?3?62?4?0?3
G The second chosen natural number multiplied by 6013179
The number with 5 hidden digits in order x45218
__________
H The product was 271903928022
The second chosen natural number6013179
Multiplied by 1526209x1526209
____________
I The product was 9177367908411
The 1st and 271903928022
2nd products +9177367908411
___________________
J Added up the sum was 9449271836433
K 5 digits were hidden 44764
The second incomplete result was 9??92?183??33
With these two incomplete results 10690?3?62?4?0?3
9??92?183??33

Can you find out the unknown figures of A. to K. ?
Each hidden digits can be 0 or 1 or 2 or 3 . . . . . or 9. There are 10 possibilities for each of them.
There are 10 hidden digits. There are 1010 = 10,000,000,000 combinations.
Of course you can try one by one all the possibilities with computer, and you can get the right answer.
Without computer you still can find them out in a short time, Can't you? 

CHALLENGE 

           100 times of a natural number minus the natural number itself, then the difference multiply by 174603.  The product is 

7 ? 9 6 3 ? 9 2 ? ? 3 6 ? 9 7 6 8 or  7 a 9 6 3 b 9 2 c d 3 6 e 9 7 6 8 .

( 5 digits are missing or hidden )

Another natural number multiply by 3123300 and the number formed by a b c d e in order (10000a + 1000b + 100c + 10d +  e )  respectively.   Add the two products up and the sum is

1 8 ? 9 8 3 ? 9 6 ? 2 ? 5 ? 3 5 6 .

( 5 digits are missing or hidden )

            What are these two natural numbers?

            Using no computer, the first one who can give me the correct answer, I will reward him or her a certificate.

In the Bible, there is a number, the number of the beast ( 666 ) is even more difficult to calculate. "This calls for wisdom: let him who has understanding reckon the number of the beast, for it is a human number, its number is six hundred and sixty-six."  Revelation

The website http://english.sdaglobal.org/ may help you to study this extremely difficult question.

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