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.

The six numbers are:

244444
424444
442444
444244
444424
444442

The total is 244442.

Given that each of the six numbers is composed of the same two digits, the six numbers must be comprised of the digits 2 and 4.
There are several combinations that will yet result in a total comprised only of these two digits, yet more than one combination is possible for that given total by swapping 2s and 4s among the six numbers. The only total where this is not true is where each of the six numbers is comprised of five 4s and one 2.

Posted by Dej Mar on 2012-08-12 11:38:57