Chapters 22 to 24 leave Israel entirely and follow a Mesopotamian diviner named Balaam son of Beor, hired by Balak of Moab to curse an approaching people. Israel does not appear as an actor in the episode at all; it is camped in the valley and never learns what is happening on the ridge above it.
The story is comic before it is anything else. Balaam sets out and his donkey stops in the road because it can see an armed messenger the professional seer cannot. He beats the animal three times, it speaks, and the man who is paid to perceive what others cannot is shown to be the least perceptive figure on the road. He then climbs to three vantage points, builds altars at each, and each time delivers a blessing instead of a curse, to the increasing expense of Balak.
Balaam is treated inconsistently across the canon, and the inconsistency is worth noticing rather than smoothing over. In these chapters he is scrupulous, saying only what he is given to say. Elsewhere in Numbers and in later books he is blamed for the seduction at Peor and remembered as a byword for a prophet who sold his gift.
He is also one of a very small number of figures from the Hebrew Bible attested outside it. A plaster inscription found at Deir Alla in the Jordan valley, written on the wall of a building, records the visions of a seer called Balaam son of Beor, in a text that is not Israelite and does not know the biblical story. Whatever else he was, he was a figure the region already had.