Find four distinct positive integers such that:

(i) The sum of all the four positive integers is 31, and:
(ii) Precisely one of four numbers is odd, and:
(iii) The only odd number is the largest and it is not prime, and:
(iv) The absolute difference between the middle two numbers is 2.

(In reply to Proving uniqueness by Steve Herman)

Steve,

"2 10 12 11 <-- odd number not largest, so no more possibilities starting with 2" should have been "2 8 10 11 <-- fail, prime"

2+10+12+11 = 35, which fails conditions (i), (iii) and -- with the assumed definition that "the middle two" refers to the two numbers in the sequence of the four positive integers ascending (or descending) -- (iv).

4 8 10 9 and 6 8 10 7 are easily discarded from consideration as are the sets of integers summing to 31 where three of the numbers are odd and-or where the difference between the two middle numbers is not two. 

Posted by Dej Mar on 2015-03-18 22:52:37