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 re: additional possibilities...LIST THEM by Charlie)

In fact, the program:

FOR a = 1 TO 2
 n(1) = a
FOR b = 1 TO 2
 n(2) = b
FOR c = 1 TO 2
 n(3) = c
FOR d = 1 TO 2
 n(4) = d
FOR e = 1 TO 2
 n(5) = e
FOR f = 1 TO 2
 n(6) = f
FOR g = 1 TO 2
 n(7) = g
FOR h = 1 TO 2
 n(8) = h
  FOR i = 1 TO 8
    IF n(i) = 1 THEN large1 = i:  ELSE large2 = i
  NEXT
  good = 1
  FOR i = 1 TO 7
   FOR j = i + 1 TO 8
     sum = i + j
     IF i + j <= 8 THEN
       IF n(i) = n(j) THEN
         IF n(i) = 1 AND sum <= large1 AND n(sum) <> 2 THEN good = 0: EXIT FOR
         IF n(i) = 2 AND sum <= large2 AND n(sum) <> 1 THEN good = 0: EXIT FOR
       END IF
     END IF
   NEXT
   IF good = 0 THEN EXIT FOR
  NEXT
  IF good THEN
    FOR i = 1 TO 8
      IF n(i) = 1 THEN PRINT i;
    NEXT
    PRINT " | ";
    FOR i = 1 TO 8
      IF n(i) = 2 THEN PRINT i;
    NEXT
    PRINT
  END IF
NEXT
NEXT
NEXT
NEXT
NEXT
NEXT
NEXT
NEXT

finds only

 1  2  4  8  |  3  5  6  7
 3  5  6  7  |  1  2  4  8

with the only difference being the order of the groups, left vs right.

Posted by Charlie on 2013-01-30 10:44:07