What is the significance of the three times that the angel hindered Bil'am?
Rashi (citing the Tanchuma): It hints at the three Avos. 1
R. Bachye (on Pasuk 23): The first time (?Vateit ha?Ason min ha?Derech? - to inform Bil?am ? that he was going on the wrong path; the second time (?Vatilchatz es Regel Bil?am?) - that he should retrace his steps; the third time (?Vatirbatz Tachas Bil?am?) - that hs should stop what he was doing. 2 The most blatant sign of all however, was the angel who told him straight - in Pasuk 32 - that the wy on which he was going was crooked and that he should retract. 3
Da'as Zekenim (on Pasuk 23), Hadar Zekenim (on Pasuk 25), Rosh and Moshav Zekenim (on Pasuk 33), citing the Tanchuma: The first time, it stopped in the field, with room on both sides - if he wants to curse the descendents of Avraham, he may do so from one side (Yishmael) or the other (B'nei Keturah), but not in the middle (Yitzchak). It next stopped in Mish'ol ha'Keramim, which has room to go two abreast. If he wants to curse B'nei Yitzchak, he may do so only from one side (Eisav), but not the other (Ya'akov). The third time there was no room on either side; he may not curse any descendents of Ya'akov, for they are all Tzadikim. Rosh - "be'Makom Tzar" hints at Ya'akov - as the Torah writes in ( Bereishis, 32:8) "Vayeitzer lo".
R. Bachye: And that was the rirst time that he acknowledged his sin, as the Torah writes in Pasuk 34.
