Keilim, Chapter 18, Mishnah 5. A basic principle runs through this masechet: a keli that breaks to the point where it can no longer do the job it was made for is no longer a keli, and it becomes tahor. Our mishnah applies that rule to a bed, which in the language of the Mishnah is built from a frame (two long sideboards and two shorter end boards) resting on four legs. The question before us is precisely how much of that bed has to be taken away before the bed that was tamei becomes tahor.
The bed and its tum'ah
The subject is announced in the mishnah's opening words: "Mitah shehaytah temei'ah midras". The bed before us has already taken on the tum'ah called midras. This is the tum'ah conveyed by a zav and by those in his halachic category to anything fashioned for lying or sitting. What sets midras apart is that touching plays no role in it at all: once the zav rests his body weight on the bed, pressing or leaning against it, the bed has become tamei. Our starting point, then, is a bed burdened with a severe tum'ah, and the mishnah asks at what stage of damage that tum'ah is finished.
Removing a short end board with its two legs
The first case: "Nitlah kitzarah ushtei kera'im, temei'ah". Should one of the short end boards be detached along with the pair of legs joined to it, the bed holds on to its tum'ah. The reason is that the remainder, the two long sideboards together with the surviving end board and its legs, is still serviceable as a bed. Granted, the end that lost its legs now dips toward the floor and lying there is hardly pleasant, yet a person is able to stretch out on it. Since the object continues to function in the way beds function, it retains the status of a bed, and the tum'ah remains upon it.
Removing a long sideboard with its two legs
The second case: "Arukah ushtei kera'im, tehorah". When what is removed is one of the long sideboards together with the two legs attached along it, the ruling reverses and the bed becomes tahor, for now we treat it as a keli that has broken. The leftover pieces, namely both end boards plus the single surviving sideboard with its legs, cannot function as a bed in any way. The entire frame keels over sideways, and whoever tried to lie there would simply slide off. Something that cannot be lain upon is not a bed, and the tum'at midras leaves it.
The view of Rebbi Nechemya
"Rebbi Nechemya metamei": Rebbi Nechemya holds that even in this second case the bed remains tamei. In his view, as long as there is still some way for the bed to serve its purpose, for example by propping it up against another bed or supporting it with a second bed, it has not lost its identity as a keli; it is still a bed and it is still usable. The Tanna Kamma disagrees: a bed must be able to stand and hold itself up on its own. Once it depends on outside support, it counts as broken, and it is tahor.
Three more ways a bed becomes tahor
Three further scenarios round out the mishnah. "Gadad shtei leshonot lochsan": he chopped away two of the small tongues of wood fitted beneath the bed's blocks, and the two he removed were the ones set diagonally, each facing the corner across from it. "Gadad shtei kera'im lochsan": alternatively, two legs standing diagonally opposite one another were cut back, with at least a tefach taken off the lower end of each. In either version the bed is robbed of all steadiness, since two of its corners now sit beneath the level of the remaining pair. "O shemi'atah pachot mitefach": and a third possibility is that he trimmed or snapped the ends of the legs until the bed rose less than a tefach above the ground, leaving it too low to be considered a bed altogether.
In all three of these cases the ruling is "tehorah": the bed is tahor from here on. Once the bed can no longer do what a bed is meant to do, hold a person lying upon it, steadily and at a normal height, it has ceased to be a keli, and with that its tum'ah comes to an end.