Taharos, Chapter 1, Mishnah 6. Our mishnah continues a discussion about foods of different levels of tumah that have been mixed together. It works through case after case, and its interest lies chiefly in what happens when such a mixture is subsequently divided up into portions.
The background
Two principles govern everything here. First, tumah descends in steps: a rishon makes what it touches a sheni, a sheni makes it a shelishi, and so on down the ladder. Not every food travels the whole length of that ladder. The level of shelishi belongs to terumah, and revi'i belongs to kodesh; ordinary chullin does not become a shelishi at all. Second, a kabeitzah, the volume of an egg, is the minimum quantity of food capable of transmitting tumah onward. Less than a kabeitzah can be tamei, but it will not pass tumah along.
A sheni and a shelishi
The mishnah's opening case sets an egg's volume of food bearing second-degree tumah, "kabeitzah ochel sheni," beside an identical measure at third degree, "kabeitzah ochel shelishi." Once the two are worked into one mass, the ruling is "shiblan zeh vazeh, sheni": the entire lump carries second-degree tumah, because there was a complete measure of sheni-level food present to draw its partner up to its own rank. Nothing here is surprising, and the mishnah records it only to lay the ground for the ruling it wants to teach next.
"Chelkan, zeh shelishi vezeh shelishi." Divide that mixture back into two portions and neither one is a sheni any longer. The food is blended, so no portion holds a full kabeitzah of sheni-level food, and each lump is therefore a shelishi.
What happens if the two portions reach terumah one at a time rather than together? The mishnah's wording is "nafal zeh le'atzmo," this one fell on its own, "vezeh le'atzmo," and that one fell on its own, each of them landing "al kikar shel terumah." The verdict is "lo pasluhu," they have not disqualified the loaf. Since every portion is no more than a shelishi, and whatever a shelishi touches could rise no higher than revi'i, and terumah has no capacity to take on tumah that far down the chain (only kodesh reaches revi'i), the loaf is left entirely as it was.
"Naflu sheneihem ke'echad, asahu shelishi." But if both portions landed at the same moment, together they once more make up a kabeitzah at the level of sheni, and a sheni renders terumah a shelishi. The loaf is now pasul and must be burned.
A rishon and a shelishi
The next case jumps over a rung of the ladder. Take "kabeitzah ochel rishon," an egg's volume of first-degree food, and set it together with an egg's volume of third-degree food, "kabeitzah ochel shelishi." Blend them, and the ruling is "shiblan zeh vazeh, rishon": the whole mass ranks as a rishon, for a complete measure of first-degree food is present and it pulls the shelishi up to its own level.
"Chelkan, zeh sheni vezeh sheni": break the mass in two and each half is a sheni. Two movements have taken place at once. The material that entered as a rishon slips down a rung, since neither half retains a whole kabeitzah of it. Meanwhile the material that entered as a shelishi has been lifted, as the mishnah spells out: "she'af hashelishi," even the third-degree food, "shenaga barishon," which had contact with the rishon, "na'asah sheni," itself turned into a sheni. Both components thus meet at the identical rank.
Larger quantities
Now the mishnah enlarges the quantities, and it is worth sketching this out. We begin with "kishtei veitzim ochel rishon," two egg-volumes of first-degree food, and place beside it two egg-volumes of second-degree food, "kishtei veitzim ochel sheni": four kabeitzim in total. Work the whole thing together and the ruling is "shiblan zeh vazeh, rishon," the entire batch carries first-degree tumah.
"Chelkan, zeh rishon vezeh rishon." Split the four kabeitzim into two portions of two kabeitzim each, and each portion is still a rishon, because each one contains a full kabeitzah's worth of the original rishon-level food.
"Lishlosha ule'arba'ah, harei eilu sheni." Divide it instead into three portions or four, and once you do the arithmetic, assuming the portions are equal, no portion holds a full kabeitzah of the original rishon material any longer. Each lump therefore falls to the level of sheni.
The mishnah now runs the identical pattern one degree lower down. Begin with two egg-volumes of second-degree food, "kishtei veitzim ochel sheni," and add an equal amount of third-degree food, "kishtei veitzim ochel shelishi." Worked into one mass, the batch is a sheni. Divide it into two: "chelkan, zeh sheni vezeh sheni," each half remains a sheni, since a full kabeitzah of the original second-degree food is present in each. Split it further, however, and the ruling changes: "lishlosha ule'arba'ah, harei eilu shelishi." Three portions or four, and no portion retains a whole kabeitzah of the original sheni material, so each one settles at the level of shelishi.
The principle in short
When the mixture contains a full shiur of the higher level of tumah, the mixture follows that higher level. When the components can only reach a shiur by combining with one another, we measure them by the lowest common denominator.