Questions & Answers about Li ne scias kiel mem solvi la problemon.
Mem is an emphasizing word meaning “(him)self / on his own / without help”.
- Without mem: Li ne scias kiel solvi la problemon. → He doesn’t know how to solve the problem (no comment about help).
- With mem: Li ne scias kiel mem solvi la problemon. → He doesn’t know how to solve the problem by himself / on his own (without others doing it for him or helping).
So mem is not grammatically required, but it adds the nuance of doing it personally, without help.
Solvi is the infinitive, corresponding to English “to solve”. After scii in this kind of structure, Esperanto uses kiel + infinitive:
- Li ne scias kiel solvi la problemon. → He doesn’t know how to solve the problem.
Using solvas (kiel li solvas la problemon) would form a full clause “how he solves the problem”, which normally describes a known, habitual way of solving it, not the idea of how to do it in principle.