Determine a list of eight positive integers (not necessarily distinct) such that summing seven of them in all eight possible ways generates only seven distinct results: 418, 420, 423, 424, 426, 428 and 429.
I started by dividing all the numbers by 7. The average of the numbers would be between 59 and 61, and the difference between the maximum and the minumum number would be:
429  418 = 11
I started with the Minimum of 60  (11/2) = 54.5.
The rest of the numbers would vary from the Minimum by
429  x
where x is one of the other distinct results. The variations were 1,3,5,6,9, and 11
54 did not work, but 55 did.
55, 56, 58, 60, 61, 64, 66 and 64

Posted by Leming
on 20060728 16:01:00 