This problem, along with the rest of the problems in the Proof for All (st)ages feature, provides an excellent context for observing, conjecturing and thinking about proof, and for appreciating the power of algebra.
This problem featured in an NRICH Secondary webinar in January 2022.
These printable cards for sorting may be useful:
Impossible Sums Proof 1
Impossible Sums Proof 2
Impossible Sums Proof 3
Impossible Sums Proof of Converse
Is there an obvious first line of the proof in each case?
Which line follows immediately from the previous line?