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**jackme12****Member**- Registered: 2012-12-17
- Posts: 6

Can you notice any patterns in the sums of the digits in three digits numbers and the sums of the digits of their complements to 1000.

For example 1000 - 255 = 745

2+5+5 = 12

7+4+5 = 16

12+16 = 28

Bonus points for multiple patterns,

Bonus bonus points for anyone who can find a direct proof.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,401

For now I only have 10 and 28...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

Let be a three-digit number.

Then

.If

, we write .Then the sum of digits is

.If

, we have .If

, the sum of digits is .If

, then and the sum of digits is .So the sums are

(i) 28 if

(ii) 19 if and

(iii) 10 if and

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

If you cast out nines on the sum of the digits of both numbers together, the cast out digit will be 1.

If c and c' represent the two complementary numbers then casting out nines on these two will

always produce tens complements. For example COD(255)=3 and COD(745)=7;

COD(111)=3 and COD(889)=7; COD(100)=1 and COD(900)=9 (9 is equivalent to 0 when casting

out nines.); COD(257)=5 and COD(743)=5.

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**mathaholic****Member**- From: Juliania
- Registered: 2012-11-29
- Posts: 2,842
- Website

1000-333=667

3+3+3=9

6+6+7=19

9+19=28

Don't know if correct, jackme12. Reply me if correct or not.

"Double the fun, double the thrill, double the coolness" - Julianthewiki

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**jackme12****Member**- Registered: 2012-12-17
- Posts: 6

Wow first time using this forum so I didn't expect an answer as thorough as scientia's. I wish I had more bonus points to give out.

Julianthemath that is the correct way to start, there are three patterns to be found depending on the initial three digit value you subtract.

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**scientia****Member**- Registered: 2009-11-13
- Posts: 222

In general, if you have an *n*-digit number

and

is theIf

is the smallest integer such that , , then , , and , .**Thus the sum of digits is **

Example:

This is equal to

(8 is the no. of digits of*Last edited by scientia (2012-12-17 22:10:50)*

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

And if casting out nines on the sum of the two complementary numbers, the result will still always

be one.

Does anyone use casting out nines anymore?

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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