Suppose that there are 6 variables: (a, b, c, d, e, f), and that some magic machine gives you the average of 5 of them any amount of times, so you can find the average of (a,b,c,d,e), or average of (a,b,c,d,f) and so on. Is it possible to solve for all values of (a,b,c,d,e,f)?
My approach to the problem:
There are 6 combinations of 5 of those variables: 6C5, they are:
(a,b,c,d,e) (a,b,c,d,f) (a,b,c,e,f), (a,b,d,e,f), (a,c,d,e,f), (b,c,d,e,f)
For each of those 6 combinations, you know the total values of each of them, so you can create a system of 6 equations:
f - e = SOME NUMBER
e - d = SOME NUMBER
d - c = SOME NUMBER
c - b = SOME NUMBER
b - a = SOME NUMBER
a - f = SOME NUMBER
However, this system of equations simplifies to still be unsolvable with infinitely many solutions. (I think it simplifies to a system of equations with 2 variables that does not create a unique solution)
Therefore, is it possible for this question to be solved? If so how, and where did I go wrong?
Thanks
NOTE: ALL VARIABLES INSERTED TO THE MACHINE MUST BE UNIQUE, WHICH MEANS YOU CAN'T HAVE (a,a,a,a,a), or (a,a,a,a,b). ITS 6C5, NOT 6P5
HINT: What do you get when you add up all the variables in {(a,b,c,d,e) (a,b,c,d,f) (a,b,c,e,f), (a,b,d,e,f), (a,c,d,e,f), (b,c,d,e,f)}?