Your question is Infer Normals From Sums. Take a moment with it on the right.
Talk me through your thinking if you like. When you're confident, submit your answer and I'll grade it like a real screen (7/10 or better passes).
Explain how you would infer normal random variables given only their sums. Keep the answer focused on the statistical method, the assumptions needed, and how you would recover the conditional distribution of the latent variables.