In this chapter, Balaam concludes with his third and final blessing, and then also issues a prophecy concerning the future.
This chapter continues directly where the last one left off, but it also adds some interesting details. It tells us that Balaam "did not go as at other times to seek omens", which clearly suggests that he was using standard divination techniques to make his previous prophecies. I had already hinted at this when I said the animal sacrifices could have been used for entrail divination, and this chapter seems to confirm it.
Even more interesting, this chapter also says that the spirit of God came upon Balaam, which is a surprisingly strong validation of the Aramite prophet. This suggests to me that previously Balaam was attempting to resist the LORD, but the LORD overpowered his attempts to curse Israel. Now it seems that Balaam "saw that it pleased the LORD to bless Israel", and he decides to allow this to happen. When he agrees to bless Israel, the LORD comes upon him in an even stronger way, and he issues his third blessing. This blessing seems largely consistent with the last two and it uses similar language, speaking of the LORD as the "horns of the wild ox" on Israel's behalf, and Israel is again compared with a lion. He concludes by repeating part of Abraham's blessing (Gen 12:3), though this is probably just a standard formula and not intended to recall that earlier blessing.
Balak responds by refusing to pay him, which I guess makes sense considering he was hired to pronounce a curse.
Balaam concludes with a different kind of prophecy. The first three prophecies were pretty straightforward "bless Israel" rants with no real predictive element. This prophecy, on the other hand, is "what this people will do... in the days to come." I am treating this fourth prophecy separately because Balak groups the first three prophecies together in v. 10 when he says "you have persisted in blessing them these three times". So while the fourth prophecy can be considering a blessing in some sense, it is different thematically (emphasizing future predictions and less of a formal blessing) and it is separated from the first three blessings by the events of verses 10-13, which serves as sort of a short conclusion to the story.
That said, the fourth prophecy is definitely the most interesting in my opinion because it contains so much more predictive content. The first prediction is that a star, or scepter, would rise out of Israel. This is evidently a reference to some emerging king, whose rule is characterized by the prophecies that follow. The "scepter" reminds me of Gen 49:10 which also talks about a scepter "not depart"ing from Judah. However, the underlying Hebrew word "shebet" is used broadly in the OT to refer to a "rod", "staff", "scepter" (in this case), or "tribe/clan". The last usage is an implied reference, as the rod refers to rulership over a tribe, so the OT will sometimes speak of the twelve "rods" of Israel. For example, see Gen 49:28, Ex 24:4 or Ex 28:21 (among others) for this usage.
The word "star" here is "kokab" and it is used less frequently in the OT to refer to both "star" literally or "prince" figuratively. We saw this second usage in Gen 37:9 when Joseph dreamed that 11 stars bowed down to him, where those stars symbolized his brothers (the princes of Israel, one could say). This same word "kokab" is used to refer to angels in other places, such as Job 38:7. All of these usages tie together in the sense of "rising power", shining and brightness. "Rising star" is an aspirational term that probably has similar meaning to the same phrase in modern English. Both of these expressions refer to a royal power rising out of Israel.
In verse 17 we see that this royal power will "crush" Moab, which is a reversal of Balak's purpose. He had tried to curse Israel, but Israel was blessed instead and now Moab is facing the prospect of defeat by Israel, which had been Balak's fear. I had mentioned back in my commentary on Num 22 that there would be a growing conflict between Moab and Israel, and that's predicted here in Num 24 as well. I feel like this conflict was one of Balak's own creation as he sought a "strike first" policy against Israel; from this prophecy we can see how badly this policy will turn out for Moab.
On a related note, v. 18 tells us that Edom will also be conquered by the rising scepter of Israel. Similar to Moab, Edom has positioned itself with a lot of hostility towards Israel, prohibiting them passage into the promised land back in Num 20.
Amalek, like Moab and Edom, resisted the Israelites and was the first nation to fight against them in Ex 17 when they were still in the Sinai. In that same chapter Israel was commanded to fight against Amalek until destruction and the fall of Amalek is reaffirmed here.
Seir is a nation that hasn't had any hostile relations with Israel yet, but they are included as a "possession" here. Both Seir and the Kenites are small nations that often fall into the orbit of the larger, powerful nations around them. I think the reference to Seir is not so much divine punishment for any wrongs they have committed as much as it's a general statement about future Israel's regional hegemony.
The reference to the Kenites, on the other hand, is a prediction of the rising power of Asshur (i.e. Assyria). Assyria has not been talked about much yet, but it will be a serious power in later times.
The final prediction here is that ships would come from Kittim (Cyprus in some translations) and conquer Asshur and Eber. Eber is not easy to identify, but from Gen 10 and 11 we can see that there was a person named Eber who was one of the descendants of Shem and a distant ancestor of Abraham. Either way, Eber is not a significant figure/nation, as he is only referenced here and in the Genesis genealogy (which is itself duplicated in 1 Chronicles). So this part of the prophecy is not biblically significant, while it's historical significance is more debatable.
So in conclusion, we can see a couple trends in this prophecy. One obvious trend is God retaliating against Israel's enemies with dire futures for Amalek, Edom and Moab. Another trend is the rising power of Israel as they become a regional force, conquering Moab, Edom and Seir, three of their neighbors. The last trend is the future rise of Assyria which will conquer the Kenites at one point but suffer from the Cypriots at another point. All of these trends will be factors in Israelite history and influence the biblical texts that we will later read.
And with that, the story of Balaam is over.