Three times Bilam went to curse Israel, and each time God caused Bilam to bless Israel rather than curse them.
It is well known that these three visions correspond to an earlier series of events - the three times Bilam's donkey strayed from the way. Three times the donkey disobeyed and angered Bilam, just as three times Bilam disobeyed and angered Balak.
Bilam was supposedly the prophetic expert who Balak looked up to, but this parallel implies that the donkey was even higher than Bilam. In the pre-Yom-Kippur prayer Tefillah Zakah we say "With the power of speech you divided mankind from animals, but I have not even been like an animal, for I have polluted my mouth with obscenities, gossip, lies..." Bilam, who had Divine gifts but used them improperly, did not even reach the level of an animal - and the story of the donkey shows this.
2. The differences between the three
One's natural assumption, when confronted with a grouping of three things, is to look for the commonalities between the three. This year, I looked instead at what differs between them.
Here are Bilam's three visions about Israel, ignoring Bilam's introductory remarks about himself:
"Behold, it is a people that shall dwell alone, and shall not be reckoned among the nations. Who hath counted the dust of Jacob, or numbered the stock of Israel? Let me die the death of the righteous, and let mine end be like his!"Note the progression in these visions. In the first, Israel dwells alone, and does not interact with the nations. In the second, Israel escapes Egypt, and will eat the flesh of its "prey". In the third blessing, Israel goes to war against its enemies, and defeats them.
"None has beheld iniquity in Jacob, nor has one seen perverseness in Israel; the LORD his God is with him, and the shouting for the King is among them. God who brought them forth out of Egypt is for them like the lofty horns of the aurochs. For there is no enchantment with Jacob, neither is there any divination with Israel; now is it said of Jacob and of Israel: 'What has God wrought!' Behold a people that rises up as a lioness, and as a lion lifts himself up; he shall not lie down until he eats of the prey, and drinks the blood of the slain."
"How goodly are your tents, O Jacob, thy dwellings, O Israel! As valleys stretched out, as gardens by the river-side; as aloes planted of the LORD, as cedars beside the waters; Water shall flow from his branches, and his seed shall be in many waters; and his king shall be higher than Agag, and his kingdom shall be exalted. God who brought him forth out of Egypt is for him like the lofty horns of the aurochs; he shall eat up the nations that are his adversaries, and shall break their bones in pieces, and pierce them through with his arrows."
Remember that the very reason Bilam came was that the Moabites feared Israel after Israel's conquest of the Amorites! Now look at what the series of visions means for Moav. In the first vision, Israel does not interact with other nations. In the second vision, Israel is violently consuming "prey" - to me this seems ambiguous, and could refer to material consumption or to defeat of enemies. The third vision explicitly has Israel defeating its enemies!
I get the impression that the more Balak insisted on a curse, the more he dug his nation into a hole. In the first vision, Israel would dwell separately and presumably not threaten anyone. In the second vision, Israel was a possible threat to others. In the third vision, Israel was a definite threat to others!
Now let's see what happens when we examine the three donkey episodes in the same way.
And the ass saw the angel of the LORD standing in the way, with his sword drawn in his hand; and the ass turned aside out of the way, and went into the field; and Balaam smote the ass, to turn her into the way.To summarize: In the first episode, the donkey was forced off the path, into a field. In the second episode, the donkey was forced into an alley, and thus tried to edge its way past the angel blocking most of the alley. In the third episode, the angel blocked the entire alley, and the donkey was forced to collapse.
Then the angel of the LORD stood in an alley between the vineyards, a fence being on this side, and a fence on that side. And the ass saw the angel of the LORD, and she thrust herself unto the wall, and crushed Balaam's foot against the wall; and he smote her again.
And the angel of the LORD went further, and stood in a narrow place, where was no way to turn either to the right hand or to the left. And the ass saw the angel of the LORD, and she lay down under Balaam; and Balaam's anger was kindled, and he smote the ass with his staff.
Once again, we see a similar progression! In the first episode, the donkey's progress was slowed, but one can take a path through the fields and get to the same destination as by a road. In the second episode, the donkey's progress was mostly blocked. In the third episode, its progress was entirely blocked. Every time Bilam failed to notice the real cause of the donkey's errant behavior, the behavior got worse. When he failed to understand a third time, the next step was for the donkey to miraculously open its mouth and for the angel appear - and only then did he understand where he had gone wrong.
In a sense, the donkey opening its mouth is a fourth episode - and it is paralleled by a fourth episode in Bilam's interactions with Balak. After Bilam's third failure to curse Israel, without prompting he begins a fourth speech: "Let me advise you what this people will do to yours in the future." This speech too is about how Israel will defeat its enemies - but unlike the previous speeches, the enemies are named, and the first one mentioned is Moav itself. Balak really should have stopped while he was ahead.
3. The three-four structure
Both the visions and the donkey episodes have what is called a "three-four" structure. This structure appears elsewhere in Tanach, most notably at the beginning of Amos. There the sins of various nations are listed:
Thus saith the LORD: For three transgressions of Gaza, yea, for four, I will not reverse it: because they carried away captive a whole captivity, to deliver them up to Edom. So will I send a fire on the wall of Gaza...This chapter contains a long list of nations which are punished, of which I have only quoted two. For each nation, the "three-four" structure is explicit. Each nation has already sinned in three ways, but once the fourth national sin is committed, the amount of sin becomes overwhelming and a punishment is decreed against the nation.
Thus saith the LORD: For three transgressions of Tyre, yea, for four, I will not reverse it: because they delivered up a whole captivity to Edom, and remembered not the brotherly covenant. So will I send a fire on the wall of Tyre...
In Bilam's case, too, we see the same three-four structures. Three times Balak wrongly asks Bilam to curse Israel; only in Bilam's fourth speech is Moav's punishment decreed. Three times Bilam ignored the donkey's behavior and hit it; only when the donkey speaks in words does the angel appear. Here, too, the sin builds up steadily, and only at the end is the verdict reached.