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Sixes and Sevens--Not! (Posted on 2012-08-12) Difficulty: 4 of 5
There are six different 6-digit positive integers that add up to a seventh 6-digit integer. Interestingly, all seven of these numbers consist of combinations of only two different digits. That is, only two different digits are used to write the complete set of seven numbers--the same two digits in each number.

So far you can't deduce what the numbers are, but if I were to tell you that seventh number, that is, the total, you'd know what the other six numbers were that made up that total.

What are the seven numbers?

From Enigma No. 1702, "All the sixes",by Ian Kay, New Scientist, 16 June 2012, page 32.

See The Solution Submitted by Charlie    
Rating: 5.0000 (2 votes)

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re: IMHO - STILL REDUNDANT | Comment 22 of 33 |
(In reply to IMHO - STILL REDUNDANT by Ady TZIDON)

You are incorrect.  If you were to read my last post properly you would see that there were multiple solution possibilities that I excluded on the basis of the statement:
"So far you can't deduce what the numbers are, but if I were to tell you that seventh number, that is, the total, you'd know what the other six numbers were that made up that total."

I'll explain this more clearly.

I get to this point in my working were I have deduced this structure:

1 _ 8 8 _ 8
1 1 1 8 1 8
1 1 1 8 8 8
1 8 1 8 1 8
1 8 1 8 8 8
1 _ 1 8 _ 1

I have also deduced in an earlier post that the second column from the right contains either 3 or 4 number 8's.  Both these solutions will satisfy the problem.

Except, if we try the case of 3 eights, we get a certain sum.  But if we are given that sum, there is no way we can deduce what the set of 6 digit numbers are because we won't know whether to put the 8 at the top and 1 at the bottom, or the 1 at the top and 8 at the bottom.  Therefore we conclude there are 4 eights in that column, thus there is only 1 possibility given the sum, that is, 8 at the top and 8 at the bottom.

Do you understand now?

I also employed this logic to fill in the 2nd column from the left.  See my previous post for details.

But if you still don't believe, here is an example.  Suppose someone said that the sum added to 818111.

Then I can pick two distinct sets that satisfy this:
1 1 8 8 1 8
1 1 1 8 1 8
1 1 1 8 8 8
1 8 1 8 1 8
1 8 1 8 8 8
1 1 1 8 8 1

OR

1 1 8 8 8 8
1 1 1 8 1 8
1 1 1 8 8 8
1 8 1 8 1 8
1 8 1 8 8 8
1 1 1 8 1 1

There are multiple such examples.  My solution is the only solution where given a sum, there is a unique set of answers.


Edited on August 13, 2012, 9:37 am

Edited on August 13, 2012, 9:41 am
  Posted by Chris, PhD on 2012-08-13 09:36:31

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