I heard that the Rama writes that cutting oneself is mutar (permitted) but the Sefer Hachinuch writes that it is against the letter of the law.
Can someone provide sources for both?
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You are speaking of the prohibition against cutting one’s self and are correct.
Everyone agrees that to cut oneself in the manner of idolaters or in grief over dead people is strictly forbidden. Many (e.g., Sefer Hachinuch, Bach YD 180:6, Shach YD 180:9, Minchat Chinuch #3) also agree that it is forbidden to cut oneself for no reason but the Rema disagrees.
Specifically, in mitzva 467, Sefer Hachinuch writes
According to that which appears [to come out] from the words of our Rabbis, may their memory be blessed, (Makkot 22) the liability of the negative commandment is only about one who gashes himself for the dead or for idolatry. But for one who gashes himself without a reason or from anger about his house that has fallen or his ship that has sunk - even though it is something extremely disgusting and ugly and forbidden - there is no liability of the negative commandment for it.
The Rema (YD 180:6) writes
Scratching and cutting for the deceased is prohibited, even outside the presence of the deceased Rema: But it is permitted to do so because of another source of pain.
Note there is a linked but separate prohibition (Chinuch 253) to tattoo onself.