What unique problem faces a market economy as it "solves" the economic problem?
The "economic problem" is the issue of what to produce, how to produce it, and for whom to produce it. This is a problem that every economic system must "solve." A market (or open) economy faces the unique problem of having to answer these questions without any particular guidance from any source.
In a command economy, the economy is run by the government. People producing goods and services know exactly what they are supposed to do because they are under orders from the government. In a traditional economy, everything runs the way it always has. People know what to do because they simply do things the way their parents did them.
In a market economy, there is no such guidance. People who are producing goods and services do not know what they should produce or how they should produce it. They have to try to do things in the best ways they can think of. They then find out from consumers whether they have done things "right." If they have, all is well, but if they have not they can end up losing money on their attempts. This is a problem that is unique to a free market.