Problem

Let \( S = {1, 2, \dots, 2013} \).
For each subset \( A = {a_1 < a_2 < \dots < a_k} \subseteq S \), define the weight of \( A \) as:

\[ w(A) = a_1 - a_2 + a_3 - a_4 + \dots + (-1)^{k+1} a_k \]

What is the sum of the weights over all subsets of \( S \)?

Insight: Pairing Trick

  • Consider subsets without the number 1.
  • There are \( 2^{2012} \) such subsets.
  • For each such subset \( S \), pair it with \( {1} \cup S \).
  • For each pair \( (S, {1} \cup S) \), the total weight is:

\[ w(S) + w({1} \cup S) = 1 \]

Example:

\[ w({2,5,8}) = 2 - 5 + 8 = 5
w({1,2,5,8}) = 1 - 2 + 5 - 8 = -4
\Rightarrow \text{Total} = 1 \]

Final Calculation

There are \( 2^{2012} \) such disjoint pairs, each contributing 1 to the total sum:

\[ \boxed{\sum w(A) = 2^{2012}} \]

Conclusion

The sum of the weights of all subsets of \( {1, 2, \dots, 2013} \) is:

\[ \boxed{2^{2012}} \]

Reference