Question 10
Let $n \ge 2$ be a given integer.
(1) Prove that one can arrange all the subsets of the set $\{1, 2, \dots, n\}$ as a sequence of subsets $A_1, A_2, \dots, A_{2^n}$, such that $|A_{i+1}| = |A_i| + 1$ or $|A_i| - 1$, where $i = 1, 2, \dots, 2^n$ and $A_{2^n+1} = A_1$.
(2) Determine, with proof, all possible values of the sum $\sum_{i=1}^{2^n} (-1)^i S(A_i)$, where $S(A_i) = \sum_{x \in A_i} x$ and $S(\emptyset) = 0$, for any subset sequence $A_1, A_2, \dots, A_{2^n}$ satisfying the condition in (1). (posed by Liang Yingde)
(1) Prove that one can arrange all the subsets of the set $\{1, 2, \dots, n\}$ as a sequence of subsets $A_1, A_2, \dots, A_{2^n}$, such that $|A_{i+1}| = |A_i| + 1$ or $|A_i| - 1$, where $i = 1, 2, \dots, 2^n$ and $A_{2^n+1} = A_1$.
(2) Determine, with proof, all possible values of the sum $\sum_{i=1}^{2^n} (-1)^i S(A_i)$, where $S(A_i) = \sum_{x \in A_i} x$ and $S(\emptyset) = 0$, for any subset sequence $A_1, A_2, \dots, A_{2^n}$ satisfying the condition in (1). (posed by Liang Yingde)