Best to avoid overdamper pianos. These are usually very old and have a more primitive action. If you lift the top lid and see a strip of wood running horizontally, blocking your view of the hammers, it is an overdamper.It also has a "birdcage" type vertical wire system in front of the action.
Beware of a sluggish action. All the hammers, whippens and dampers pivot on tiny center pins. These can become tight and sluggish. Gently push groups of hammers forward and see that they move back freely. If you are seeing hammers that are staying forward it probably has tight center pins. If there are several it may require an action overhaul.
Check soundboard and bridges for cracks.
Tuning pins maybe loose. Sometimes you can see they have been pulled down so there is a gap above the tuning pin. This can render the piano untunable.
Check the wear on hammers. Are there deep lines cut into the hammers from hitting the strings? The hammers can be easily reshaped if the wear is moderate.
Check there is not too much sideways wobble in the keys. The keys shouldn't rattle or click.
Is the touch responsive? Not too heavy or light. Not sluggish.
Check general condition inside. Is it clean and tidy?
This is a basic checklist. Ideally call me for a pre-purchase inspection.
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