Keilim, Chapter 18, Mishnah 9. This chapter has been occupied with the bed and its parts: when the bed as a whole is considered a keli that receives tumah, and how its individual pieces are treated. Our mishnah takes up the case of a bed that had already become tamei and then came apart, and it closes with a dispute about a leg that is still joined to one of the boards of the bed.
A bed that broke in two
The mishnah opens: "Mitah shenignav chetzyah o avad chetzyah, o chalakuha achin o shutafin, tehorah." A bed that was tamei broke into two halves, and then one of the halves was stolen, or one of the halves was lost, or the bed was divided between brothers who received it as part of their father's estate, or between partners who had owned it together, each one taking his half. In all of these cases the mishnah rules: tehorah, the bed is tahor.
Why does the tumah leave? Because a single half of a bed serves no purpose by itself. It is not a usable keli, and in the situations the mishnah describes there is no prospect of the two halves being brought back together: one half is gone, or the halves now belong to two different owners. The bed is simply broken. Once that is the case, both halves come out of their tumah and are tahor.
The mishnah continues: "Hechziruha, mekabelet tumah mikan ulehaba." Should the two halves be reassembled and the bed put back together, it resumes its status as a keli capable of receiving tumah, though only going forward. Its earlier tumah is gone for good and does not revive; from here on it is subject to fresh tumah, should it touch a tamei object.
A leg and the board attached to it
The mishnah now turns to a piece of the bed rather than the whole bed: a single leg together with one of the end boards of the bed, the two of them still attached to one another. Two questions arise. First, if the leg becomes tamei, does the board attached to it become tamei along with it? Second, when the pair is immersed in a mikveh, does the immersion work while they are still joined?
Rabbi Eliezer's position is: "mitamea chavilah umitaheret chavilah." So long as these two pieces are fastened together, halachah regards them as a single bundle, one item. Should tumah come upon the leg, the attached board is drawn into that tumah with it. The same principle governs their purification: since they count as one entity for becoming tamei, they count as one entity for becoming tahor, and dipping the joined pair in a mikveh renders both of them tahor without separating them first.
The Chachamim rule otherwise: "mitamea eivarim umitaheret eivarim." A bed in one piece is indeed a single unit, but once it has come apart, halachah looks at what remains as detached components, each limb on its own. On this view the two pieces before us are two distinct objects. Tumah that reaches the leg leaves the board untouched, and it stays tahor. So too in reverse: purification is achieved piece by piece, and dipping the leg while the board is still fixed to it will not render the leg tahor.
The reason is a practical one. The leg is fitted tightly into the board, and when the two are dipped together the water of the mikveh cannot reach into the space where they meet. Part of the surface of the tamei leg would never be touched by the water of the mikveh, and immersion requires that the water reach the entire keli. According to the Chachamim, then, the only way to purify these pieces is to take them apart from one another and immerse each one separately.