Yoma, Chapter Six, Mishnah Eight
This mishnah takes us back to the Temple, where the high priest was waiting to hear if the goat had reached the wilderness and Israels sins had been forgiven. The mishnah
They said to the high priest: the goat has reached the wilderness.
And how did they know that the goat had reached the wilderness?
1) They used to set up scouts and they would wave scarves, and they would know that the goat had reached the wilderness.
2) Rabbi Judah said: but did they not have a great sign? From Jerusalem to Bet Hidudo was three mils. They could walk a mil, return the mil, then wait the time it takes to walk a mil, and thus know that the goat had reached the wilderness.
3) Rabbi Ishmael said: but did they not have another sign! A thread of crimson wool was tied to the door of the Temple, and when the goat reached the wilderness the thread turned white, as it is written, Though your sins be as scarlet they shall be as white as snow (Isaiah 1:18).
Section one: According to the first opinion in the mishnah, they knew when the goat reached the wilderness through scouts who would signal back to Jerusalem with scarves that the goat had been pushed off of Zuk by the priests who accompanied him.
Section two: Rabbi Judah says that they didnt need to rely on scouts; rather they relied on a system of timing how long it would take the goat to get to the wilderness. Bet Hidudo was the beginning of the wilderness and once the goat had reached this point Rabbi Judah considered the mitzvah of sending the goat to the wilderness to have been fulfilled. Since Bet Hidudo was only three mils from Jerusalem, all they needed to do was time how long it would take to walk three mils. What they would do is walk a mil (which is to the end of the Shabbat border limit), then turn around and walk back the mil and then wait the amount of time it would take to walk a mil.
Section three: Rabbi Ishmael says that there is a more symbolic, perhaps even magical way of knowing when the goat had reached the wilderness. They would tie a crimson thread (similar to the thread tied to the goats horns) to the opening of the Temple and when the goat would reach the wilderness the thread would turn white in a literal (perhaps overly literal) fulfillment of the verse from Isaiah in which God promises the people of Israel that he will turn their sins white as snow. We should note that good manuscripts of the mishnah do not contain Rabbi Ishmaels words; it is likely that this is an addition to the Mishnah from the Talmud.