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.

If I tally 6 digits all of the same value only "1" yields a single digit sum.
If I have any mix of 2 digits forming those 6 I cannot have a single digit sum which is either of those.

I am told not 6's and 7's but neither are individually ruled out.

I offer:
   1 1 1 1 1 1
   1 1 1 1 1 1
   1 1 1 1 1 1
   1 1 1 1 1 1
   1 1 1 1 1 1
   1 1 1 1 1 1
   6 6 6 6 6 6

With my subject title in mind, and no disallowance of leading zeroes, I offer:
  1 0 0 0 0 0
  0 1 0 0 0 0
  0 0 1 0 0 0
  0 0 0 1 0 0
  0 0 0 0 1 0
  0 0 0 0 0 1
  1 1 1 1 1 1

Or for that matter let x be any digit other than 0:
  x 0 0 0 0 0
  0 x 0 0 0 0
  0 0 x 0 0 0
  0 0 0 x 0 0
  0 0 0 0 x 0
  0 0 0 0 0 x
  x x x x x x

I'm am sure however that the first was the intended.

The above was posted prior to an edit which inserted "different" into the text ---
"There are six different 6-digit positive integers...."

Even without the word "difFerent" I do see that I did make an assumption only after testing a few examples.

Edited on August 13, 2012, 12:45 am

Posted by brianjn on 2012-08-12 21:05:11