Question 94
Prove that given any two permutations $\sigma, \tau \in S_n$, there exists some function $f : \{1, \dots, n\} \to \{-1, 1\}$ such that we simultaneously have, for any indices $1 \le i \le j \le n$,
$$
\left| \sum_{k=i}^{j} f(\sigma(k)) \right| \le 2 \quad \text{and} \quad \left| \sum_{k=i}^{j} f(\tau(k)) \right| \le 2.
$$
Dan Schwarz
$$
\left| \sum_{k=i}^{j} f(\sigma(k)) \right| \le 2 \quad \text{and} \quad \left| \sum_{k=i}^{j} f(\tau(k)) \right| \le 2.
$$
Dan Schwarz