Material Not Behavioural

Even though 'solve' does not project, viewed 'from above', in terms of the meaning being realised, problem-solving is a cognitive mental process: coming to understand a semiotic phenomenon.

they	solved	the problem
Senser	Process: mental	Phenomenon

On the other hand, viewed 'from roundabout', such clauses can accept a Beneficiary — eg for the teacher — which is a participant in material, verbal and relational clauses only (Halliday & Matthiessen 2004: 293), thereby ruling out mental, behavioural and existential clauses.  Since the clause is clearly neither verbal nor relational, on this criterion, the clause is material.

they	solved	the problem
Actor	Process: material	Goal

So the clause combines features of mental and material processes, depending on the stratum of content — the level of symbolic abstraction — meaning or wording, it is viewed from.

Note that this is anomalously analysed as 'behavioural' in Deploying Functional Grammar (Martin, Matthiessen & Painter 2010: 125-6):

they	solved	the problem
Behaver	Process: behavioural	Phenomenon

While it is true that behavioural processes display aspects of both mental and material processes, the tension here is across strata, between semantics and lexicogrammar, not within lexicogrammar.