Simplified problem:
I have to create a set of $6$ numbers. From this set an external entity will be choosing $3$ numbers at random (repetition allowed) $-$ and compute its average, median, standard-deviation and tell these $3$ metrics back to me.
Q$1$) How will I create a set, such that, given average, median, standard-deviation, I can infer the $3$ elements which are part of the selection.
HINT: You don't need average, median and std dev. All you need is the average, because from that you can calculate the sum of the $3$ numbers.
Can you find a set of $6$ numbers s.t. the sum of any $3$ of them will uniquely identify which $3$ (with repetitions) are chosen?