Keilim, Chapter 22, Mishnah 5. This chapter is occupied with the furniture of the home, tables and chairs and benches, and with the question of what happens to their status of tumah when a piece of them breaks off or is taken away. Our mishnah takes up one particular case of a chair that has lost part of its seat.
To follow the case we first have to picture the chair as it was built in the days of the Mishnah. Its legs were not slender poles the way we imagine legs today. They were wide boards, and these boards stood along three sides of the chair, forming something closer to an open box. Across the top of those three boards the craftsman would lay three thin wooden strips, slats, and those slats together made up the surface a person sat upon. Sometimes the slats were cut long, so that their ends stuck out beyond the sides of the chair. Sometimes they were cut exactly to size, so that their ends sat flush with the boards beneath them. That small difference in carpentry is what our mishnah turns on.
Here is the ruling in the mishnah's own words: "kisei shelo hayu chipuyav yotze'in venitlu", meaning a chair built with slats that ended flush with its sides, from which several of those slats have since been removed. Once part of the seating surface is gone, nobody can sit properly on top of the chair in the manner for which it was fashioned, and we would assume that it has stopped being a vessel altogether. The mishnah's verdict runs the other way: "tamei", the chair still receives tumah.
Why? The mishnah explains, "shekein darko liheyos", such is the ordinary practice with a chair of this kind, "matehu al tzido veyoshev alav": one leans it over and makes a seat out of one of its flanks. Remember that the frame is really an open box formed by three broad boards. When the top gives way, the owner does not discard the chair; he stands it on one of those boards and seats himself there. The object thus goes on serving exactly the role for which it was manufactured, namely seating a person, and as long as a utensil continues to do its job it retains its capacity to become tamei.
Now we can appreciate why the mishnah was careful to describe the slats as not protruding. Precisely because the slats were cut flush with the sides, nothing gets in the way when the chair is laid down, and it rests steadily on its side so that a person can sit upon it. Had the slats stuck out past the sides, their protruding ends would prevent the chair from being set down that way, and there would be no remaining way to use it as a seat. It is the flush edge, that quiet detail of the carpenter's work, that keeps this chair in service and keeps it susceptible to tumah.