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 Sixes and Sevens--Not! (Posted on 2012-08-12)
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.

 Submitted by Charlie Rating: 5.0000 (2 votes) Solution: (Hide) From various considerations, only the digits 1 and 8 will work. As posted by correspondent Ogee on the NewEnigma web site (registration on the site required to view this link), the following are the only possibilities for consideration, with the addends in ascending order within their respective columns: ```111818 111818 111811 111818 111818 111818 111818 111818 111811 111881 111881 111818 111888 111881 111888 111888 111881 111818 111888 111888 111888 118888 111888 118818 118888 111888 111888 118888 118818 118888 181818 181818 181818 181811 181818 181818 181818 181818 181818 181881 181888 181881 181818 181888 181888 181888 181888 181888 181888 188888 181888 181888 188818 188888 ------ ------ ------ ------ ------ ------ ------ ------ ------ 818181 818111 818111 888181 888181 888111 888111 888111 888111 ``` Of the sums, only the first is unique. Thus the first column represents the wanted summation.

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