[Big-Daddy]

But you have also assumed in setting up the equations that there is no H⁺ or H₂O

I didn't assume anything, I just checked it works.

If we had to involve these

What for? Reaction is already balanced.

Oh. So you just used the algebraic method to check that this is the only solution?

To originally find the reaction equation we have to keep an open mind that H⁺ and H₂O could be involved (even if finally it turns out that neither is involved). To me this implies some kind of redox method of balancing. So, how did you initially find the reaction equation? As far as I can see we have 3 half-equations

S₂O₃^2- + 5 H₂O  2 SO₄^2- + 10 H⁺ + 8e^-
3 S₂O₃^2- + 3 H₂O  2 S₃O₆^2- + 6 H⁺ + 8e^-
O₂ + 4 H⁺ + 4e^-  2 H₂O

If we consider the possibility (pre-balancing) that H⁺ and H₂O might be involved in the final reaction equation, there is no way we could skip straight to that final reaction equation, since there will be infinite solutions. So what was the redox balancing method for this question?

When I originally did, I summed up the two oxidations, multiplied the reduction by 4 and then summed. Why? Just because it is the same method as on that wrong half-equations page linked above. It worked this time. But the logic behind this particular choice of coefficients is not apparent - why not multiply the first oxidation by 3, add on the second oxidation and then multiply the reduction by 8, for a random example?

[Borek]

S₂O₃^2- + 5 H₂O  2 SO₄^2- + 10 H⁺ + 8e^-
3 S₂O₃^2- + 3 H₂O  2 S₃O₆^2- + 6 H⁺ + 8e^-
O₂ + 4 H⁺ + 4e^-  2 H₂O

This is a dangerous approach. Once you have three equations like the ones you write (call them A, B and C) you can produce infinitely many linear combinations of them that will yield a "correct" answer. Just by the look at it every m(A+2C)+n(B+2C) should be "correctly" balanced (m,n being natural numbers).

Thus, the safest approach is to not split reduction nor oxidation into several half reactions.

In general there doesn't exist a truly systematic method that would allow balancing of every reaction. What is really happening sometimes depends on the mechanism details, and it can't be predicted just by looking at the formulas of the substances present.

What if I will tell you to balance

C₇H₆O₃ + C₄H₆O₃ C₉H₈O₄ + H₂O

What if I will tell you it is a synthesis of aspirin from the salicylic acid and acetic anhydride?

[Big-Daddy]

This is a dangerous approach. Once you have three equations like the ones you write (call them A, B and C) you can produce infinitely many linear combinations of them that will yield a "correct" answer. Just by the look at it every m(A+2C)+n(B+2C) should be "correctly" balanced (m,n being natural numbers).

Thus, the safest approach is to not split reduction nor oxidation into several half reactions.

Then how do you write one oxidation half-reaction for the case at hand?

I can see that not all reactions can be balanced without knowing the chemical groups involved or mechanisms. But I would like to understand this one, because at least it should help me understand disproportionations like this (where the disproportionation itself needs a reducing or oxidizing agent to enable it).

Can we use some logic to do with sulphur oxidation numbers maybe to find the overall oxidation half-reaction?

What if I will tell you to balance

C₇H₆O₃ + C₄H₆O₃ C₉H₈O₄ + H₂O

What if I will tell you it is a synthesis of aspirin from the salicylic acid and acetic anhydride?

Shouldn't ethanoic acid be the second product rather than water? In any case, I see the point you're making.