Learn Hardcore Esperanto: Li ne scias kiel mem solvi la problemon. - He does not know how to solve the problem himself.

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.

Why is the verb solvi and not solvas?

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.

Why is kiel used here? Could it be something like kio or ke instead?

Kiel means “how”, and in this sentence it introduces an indirect question: “how (to) solve the problem”.

kiel solvi → how to solve
kion fari → what to do
kie trovi ĝin → where to find it

You cannot use ke here, because ke means “that” (it introduces a statement, not a question):

Li scias, ke li solvos la problemon. → He knows that he will solve the problem.

Here we need “how”, so kiel is the right word.

Why does problemon end in -n? Why not just problemo?

The -n ending marks the accusative in Esperanto, which is used for the direct object of a verb.

The verb is solvi (to solve).
What is being solved? la problemon.

So la problemon must have -n to show it is the thing being solved:

Mi vidas la problemon. → I see the problem.
Ni solvas la taskon. → We solve the task.

Without -n, la problemo would not be in the usual direct‑object form.

Why is there la before problemon? Could I say solvi problemon?

La is the definite article, similar to English “the”.

la problemon → the problem (some specific, known problem)
problemon (without la) → a problem / some problem (more general or indefinite)

Depending on context, both are possible:

Li ne scias kiel solvi la problemon. → some particular problem already known from context.
Li ne scias kiel solvi problemon. → he doesn’t know how to solve a problem (of that type), more general.

Where does mem belong in the sentence? Can its position change?

Mem normally goes right next to the word it emphasizes. In your sentence it emphasizes the (understood) subject of solvi, so it sits just before solvi:

Li ne scias kiel mem solvi la problemon.

Other positions change what is emphasized:

Li mem ne scias kiel solvi la problemon. → He himself doesn’t know (others might know).
Li ne scias kiel solvi la problemon mem. → He doesn’t know how to solve the problem itself (or: without delegating it), often interpreted as solving that problem oneself.

So yes, it can move, but its position changes the focus of the emphasis.

If mem refers to li, why don’t we see li mem after kiel?

In kiel mem solvi la problemon, the subject of solvi is understood to be the same li from the main clause. Esperanto does not repeat that pronoun:

Literally: He doesn’t know how (he) himself to solve the problem.

Mem is not a full pronoun by itself; it is an intensifier attached to an explicit or implied pronoun. Here, the pronoun (li) is implied by the structure, so mem can stand there alone and still clearly point back to li.

Could I say Li ne scias kiel solvi la problemon mem instead? What is the difference?

Yes, that is grammatically correct:

Li ne scias kiel mem solvi la problemon. → Emphasis on himself doing the solving.
Li ne scias kiel solvi la problemon mem. → Emphasis more on solving that problem personally / by himself.

In practice, many speakers would understand both as “doesn’t know how to solve the problem by himself”, but mem right before solvi more clearly targets the doer (he), and mem after problemon more clearly targets the specific problem being done personally.

Why is it scias and not konas? Aren’t both “to know”?

Esperanto distinguishes:

scii → to know a fact, information, how to do something
koni → to know a person, place, work, to be familiar with something

Here we are talking about knowing how to do something, so we must use scii:

Li ne scias kiel solvi la problemon. → He doesn’t know how to solve the problem.

Examples:

Mi konas lin. → I know him.
Mi scias, kie li loĝas. → I know where he lives.
Ŝi scias naĝi. → She knows how to swim.

Is a comma needed before kiel, like Li ne scias, kiel mem solvi la problemon?

In Esperanto, many writers do put a comma before a clause introduced by a question word (like kiel, kial, kie, kio):

Li ne scias, kiel mem solvi la problemon.

In a short sentence like this, omitting the comma is also common, especially when kiel + infinitive is felt as a compact unit (kiel solvi = how to solve).

So both spellings are acceptable in practice, but adding the comma is slightly more in line with traditional punctuation rules for subordinate clauses.

How would I change the sentence to say “He himself doesn’t know how to solve the problem” (emphasis on he, not on by himself)?

You move mem next to li, the word you want to emphasize:

Li mem ne scias kiel solvi la problemon.

Now the main meaning is that he himself lacks the knowledge (maybe others do know), rather than focusing on whether he would do it with or without help.