Reconstructing four integers from unordered block sums: a divisibility phenomenon at 42

A hidden vector w = (w1,...,w4) of positive integers with known sum n is probed by experiments: an experiment is a set partition π of the index set {1,2,3,4}, and it returns the unordered multiset of block sums of w. We ask for the least number Q(n,4) of experiments, chosen in advance, that determine w exactly (coordinate by coordinate) among all compositions of n. We prove that Q(n,4) ≤4 for every n, and that for every n≥6 Q(n,4) = 4 if 42 |n, 3 otherwise, while Q(5,4) = 2. The upper bound for 42 ∤ ncomes from three explicit triples of experiments, each of which is injective exactly when n is odd, when 3 ∤ n, or when 7 ∤ n, respectively; the factorisation 42 = 2·3·7 is thus explained. The classification of collisions is carried out by exact linear algebra over Q combined with rational certificates of infeasibility (Farkas’ lemma). The proof is computer-assisted; all certificates are provided as ancillary files. We also report computations of Q(n,k) for k = 3 and k = 5 and state open questions. The problem is motivated heuristically by block structures in iterations of the Collatz map; no result on the Collatz conjecture is claimed.