After the Exodus, and prior to the conquest of Sihon and Og, Israel attempted to pass through Edom. (The Edomites were the descendents of Esau.) Numbers 20:18-21 records Edom's response as follows:
Edom, however, said to him, “You shall not pass through us, or I will come out with the sword against you.” Again, the sons of Israel said to him, “We will go up by the highway, and if I and my livestock do drink any of your water, then I will pay its price. Let me only pass through on my feet, nothing else.” But he said, “You shall not pass through.” And Edom came out against him with a heavy force and with a strong hand. Thus Edom refused to allow Israel to pass through his territory; so Israel turned away from him.
In Deuteronomy, Moses recounts Israel's wanderings after the Exodus, and begins to describe the conquest of Sihon and Og in 2:24. Beginning in verse 26 Moses recalls a conversation with Sihon:
“So I sent messengers from the wilderness of Kedemoth to Sihon king of Heshbon with words of peace, saying, ‘Let me pass through your land, I will travel only on the highway; I will not turn aside to the right or to the left. You will sell me food for money so that I may eat, and give me water for money so that I may drink, only let me pass through on foot, just as the sons of Esau who live in Seir and the Moabites who live in Ar did for me, until I cross over the Jordan into the land which the Lord our God is giving to us.’"
It seems like Numbers is saying the Edomites did not allow Israel to pass through, but in Deuteronomy, Moses recalls that he told Sihon that they did allow them to pass through.
How do the experts typically reconcile this apparent contradiction? Did Moses lie to Sihon? Is this describing two different groups of people? Or two different occasions? Or am I missing something in the wording of the Hebrew?
To clarify, I am asking how this is normally reconciled. I have been interpreting Scripture for long enough to know better than to assume that this is an actual contradiction. So, I am only interested in answers from the perspective that the two passages can actually be reconciled.