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Let $p\equiv 1 \pmod{4}$ be a prime. Write $t = \prod_{x=1}^{(p-1)/2}x$.Since $t ^2\equiv -1 \pmod{p}$ , we can divide $\{1,2,\ldots,(p-1)/2\}$ into$(p-1)/4$ ordered pairs so that each pair, say $<a,\tilde{a}>$ , satisfies that$t a \equiv \pm \tilde{a} \pmod{p}.$ For any two such pairs, assume$a<\tilde{a}, b<\tilde{b}, a<b $, then there are three possibilities for theirrelative order : $a<\tilde{a} < b< \tilde{b}$ , $a< b < \tilde{a} < \tilde{b}$, $a< b < \tilde{b}< \tilde{a}$. We show this paring is balanced in the sensethat the three cases occur with equal frequencies. Utilizing properties of thisparing we solve one problem raised by Zhi-Wei Sun concerning the sign ofpermutation related to quadratic residues.

Balanced paring of $\{1,2,\ldots,(p-1)/2\}$ for $p\equiv 1 \pmod{4}$