Consider Groups I and II consisting of positive integers from 1 to 7, where:

Group I = {1,2,4,7} and, Group II = {3,5,6}

It is observed that the sum of any two numbers of either group is either greater than the largest number or is in the other group.

For example, 1 and 2 belong to the first group whose sum (3) belongs to the second group. Also, 3 and 5 belong to the second group, whose sum (8) is greater than the largest number (6) in the second group.

Can you come up with arrangement of the first eight positive integers, that is from 1 to 8 inclusively, that satisfies the same two properties?

(In reply to additional possibilities...LIST THEM by Ady TZIDON)

Ady,

your (1 2), (3 4 5 6 7  8)

to take just one example, 3+4 = 7, which is not greater than the 8 that's the highest in its group, nor is it a member of the other group. This is the case for all the sums of the smallest two in the right hand group in all the cases you give.

Posted by Charlie on 2013-01-30 09:49:51