Gemara

(Rav Chisda): On Shabbos, one may take rocks up to the roof to clean himself after eliminating. Kevod ha'Briyos (avoiding disgrace) overrides a Lav (all mid'Rabanan laws, which are based on "Lo Sasur", do not veer from Chachamim's words).

Question (Ravina - Beraisa - R. Eliezer): One may take a chip of wood to remove food between his teeth;

Chachamim permit only chips in the feeding trough [suitable for animals. One may not use Muktzeh!]

Answer: A person plans where he will eat (he should prepare chips before Shabbos), but he cannot always plan where he will eliminate.

146b (Mishnah): One may break a barrel to eat the dry figs inside, as long as he does not intend to make a Kli (i.e. a nice opening).

Beitzah 33a (Mishnah - R. Eliezer): One may take a chip from in front of him to clean his teeth.

(Rav Yehudah): Tikun Kli does not apply to animal food.

Question (Rav Kahana - Beraisa): One may move fragrant wood to smell it or wave it for a Choleh. He may roll it to smell it, but he may not cut it to smell it. If he cut it, he is exempt, but it was forbidden. He may not cut it to clean his teeth. If he cut it, he is Chayav Chatas.

Answer (Rav Yehudah): The Beraisa discusses hard wood. (I discussed soft wood.)

Question: One does not roll hard wood!

Answer: The Beraisa is abbreviated. It means as follows. One may roll or cut it to smell it, i.e. soft wood. One may not cut hard wood. If he cut it, he is exempt. If he cut it to clean his teeth, he is Chayav Chatas.

(Beraisa #1): He may cut it to smell it.

Contradiction (Beraisa #2): He may not cut it to smell it.

Answer #1 (Rav Chisda): Beraisa #1 discusses soft wood, and Beraisa #1 discusses hard wood.

Objection #1 (Rav Acha bar Yakov): Why is hard wood forbidden?

(Mishnah): One may break a barrel to eat the figs inside...

Objection #2: Rava bar Rav Ada said that in Rav Yehudah's house, he used to cut and give us give big sticks, even though they could be used for handles for axes and chisels.

Answer #2: Rather, R. Eliezer forbids and Chachamim permit;

Beraisa (R. Eliezer): One may take a chip from in front of him to clean his teeth;

Chachamim say, one may take only from an animal's trough.

Both agree that one may not cut it.

R. Eliezer says, if he cut it on Shabbos to clean his teeth or open the door, he is Chayav Chatas;

Chachamim forbid only due to Shevus.

There R. Eliezer is Mechayav Chatas, so here he forbids l'Chatchilah. There, Chachamim forbid only due to Shevus, so here they permit.

Question: Does R. Eliezer argue with the Mishnah that permits breaking a barrel?

Answer (Rav Ashi): That refers to a broken barrel stuck together with sap. (We are not concerned lest he intend to make it a Kli.)

Rishonim

Rif (Beitzah 19a): Tikun Kli does not apply to animal food. One may take a chip to clean his teeth. One may roll and break fragrant wood to smell it, whether it is hard or soft, just like one may break open a barrel to eat the figs, as long as he does not intend to make it a Kli.

Rambam (Hilchos Shabbos 11:7): If one takes a chip from in front of him and cuts it to clean his teeth or to open the door, he is liable.

Magid Mishneh: Based on the Sugya, I do not understand why the Rambam says that one is liable. Chachamim say that cutting in order to clean the teeth is only Shevus. Do not say that this is only from a trough, which is soft, but they agree about hard chips. We hold like Rav Yehudah, who used to cut hard sticks for smelling. It seems that even if one cuts for a Kli, he is exempt. When the Gemara challenged Rav Yehudah, it could have answered that the Beraisa is R. Eliezer, and the Halachah does not follow him, for he forbids cutting to smell. The Gemara answered the truth, that even R. Eliezer obligates only for hard wood. The second Beraisa discusses hard wood. Rashi and the Rashba explain like this. Rashi says that Chachamim say that it is only Shevus, for it is Tikun k'Le'acher Yad (unskillfully). It is a proper Tikun only if he uses a knife. If the Rambam discusses cutting with a knife, he should have specified. I did not find a nice explanation for the Rambam. It seems that the Rif explains like Rashi.

Rambam (8): Anything proper for animal food, e.g. straw and moist herbs, one may cut it on Shabbos, for Tikun Kli does not apply to it. One may cut fragrant wood to smell it, even if it is hard and dry. One may break off whatever he wants, be it a big branch or a small branch.

Rosh (Beitzah 4:13): Rashi explains that R. Eliezer and Chachamim agree that one may not cut a chip. The Gemara connotes like this, for [something from] an animal's trough is not suitable to open a door. Chachamim forbid only due to Shevus when he cut with his hand, which is unskillful. If he cut with a knife to clean his teeth or open a door, why do they exempt? The Tosefta connotes that 'they agree' refers to from the trough. It teaches only R. Eliezer's opinion, regarding from a trough. According to Rav Yehudah, who says that Tikun Kli does not apply to animal food, we can say that this is when he cuts by hand, and the Tosefta discusses with a knife. Our Gemara disagrees. Rabanan say that if he cut a chip to clean his teeth, with a knife he is Chayav Chatas, and by hand it is Shevus. From a trough one may cut even with a Kli, for cleaning his teeth. Only regarding a chip, for which there is a Chiyuv Chatas with a Kli, we decree to forbid by hand due to with a Kli. We do not decree about using a Kli to cut something from a trough due to using a Kli to cut a chip. A support is that the Gemara compares this to breaking a barrel, which is with a Kli. However, the Tosefta connotes that R. Eliezer discusses something from a trough, and Chachamim forbid using a Kli to cut from a trough. Our Gemara is primary. Rav Yehudah taught that Tikun Kli does not apply to animal food, i.e. one may cut a chip from there to clean his teeth. The Halachah follows this.

Poskim

Shulchan Aruch (OC 322:4): Tikun Kli does not apply to animal food. Therefore one may cut straw even with a knife to clean his teeth.

Gra: The Gemara compared this to breaking a barrel. See 314:6 (which permits slicing off the top of a barrel, but not to puncture it).

Mishnah Berurah (9): The Gemara connotes that something proper for animals to eat, even if it is not destined for them, e.g. soft wood that they could eat, is not called a Kli.

Mishnah Berurah (11): Some say that there is no Tikun Kli only when he makes a Kli from the beginning, but if he fixes an existing Kli, e.g. there was a Kli with a hole, and he took a piece of turnip and cut it to the size of the hole, Tikun Kli applies (Chayei Adam citing the Ran. He holds that one is Chayav Chatas for this.)

Mishnah Berurah (12): Even if he was particular about the size, there is no concern for Mechatech (cutting to size), since it is a food.

Bi'ur Halachah (DH Ochlei): The Shulchan Aruch is difficult, for based on 308:30, it should be forbidden to move animal food. (It says there that even where people feed date pits to animals, an important person should move them only through a Shinuy.)

Shulchan Aruch (ibid.): A chip that is not animal food, one may not even take it to clean his teeth.

Beis Yosef (DH Aval): Rashi explains that Chachamim permit taking only from a trough, for what is proper to eat is prepared for all needs. Wood is usually used for something forbidden, e.g. making Kelim, therefore one may move it only for a fire, i.e. on Yom Tov. On Shabbos it is forbidden in every case. The Halachah follows Chachamim.

Magen Avraham (4): If one cut in order to clean his teeth, he is liable.

Gra (4): This is like the Rambam. He holds that 'all agree...' refers to from a trough. This is like the Tosefta. The Rosh and Shulchan Aruch are unlike the Tosefta and Rambam. Why did the Magen Avraham bring from the Rambam that one is liable? We do not hold like him!

Magen Avraham (4): The Gemara connotes that if one was invited to a meal and he could not prepare before Shabbos something to clean his teeth with, he may move a chip to clean his teeth if he has nothing else, due to Kevod ha'Briyos. It is disgraceful, for sometimes a piece of meat between the teeth is visible outside. See 312:1 (which permits using rocks to clean oneself on Shabbos.) Where this is not considered disgraceful, it is forbidden.

Kaf ha'Chayim (15): Tosefes Shabbos says that if we know what most people in the world do, we do not follow the local custom.

Gra (4): There are three laws. One may move and cut animal food, even to clean his teeth. One may move and cut fragrant wood to smell; to clean his teeth is exempt, but forbidden. One is exempt also for a chip, but one may not even move a chip.

Mishnah Berurah (13): It is Muktzeh like Stam wood, so one may not take it even to use without cutting it.

Mishnah Berurah (14): Even though it can be an issue of Kevod ha'Briyos, it is forbidden, for he should have prepared before Shabbos.

Kaf ha'Chayim (16): One may use a chip that he prepared, even if it is very weak and surely the end will break through cleaning his teeth.