ClO₃^–(aq) → ClO₄^–(aq) + ClO₂^–(aq)

basic environment

ClO₃^–(aq) + H₂O(l) → ClO₄^–(aq) + 2 H⁺(aq) + 2 e^–

ClO₃^–(aq) + 2 H⁺(aq) + 2 e^– → ClO₂^–(aq) + H₂O(l)

sum and eliminate common species

2 ClO₃^–(aq) → ClO₄^–(aq) + ClO₂^–(aq)

This one could easily be balanced by inspection.