The Red Heifer
The red heifer was to atone for the Golden Calf. The medrash states, “...let the mother (Red Heifer) come and atone for the child (Golden Calf). Let the one who is Red come and atone for the one who is Gold.”
It is clear that some form of atonement is effectuated for the sin of the Golden Calf via this Red Heifer. But atonement must be defined first. Atonement is normally when one commits never to commit a specific evil act or character trait again. How then can an external act of offering the Red Heifer atone for the sin of the Calf?
G-d then forgives the person for his past crimes, as the person does not identify himself with that “old me”, and G-d does not consider prior sins as having been committed by this “new” person. He is completely forgiven. A true act of kindness by G-d.
I believe the Red Heifer works in the same way. The reason why the Jews sinned with the Golden Calf is because they were accustomed to this species. It was present in Egypt. Their familiarity with this animal allowed them to express their idolatrous wishes. To remove the possibility of the Jew committing such a sin again, G-d created the Red Heifer as a means of removing one's familiarity with this species. How does this work? The answer may be that by creating self contradictory laws in the Red Heifer (it purifies one who is ritually impure, and contaminates one who is pure) one is distanced from a simple understanding of this aspect of the Red Heifer. Meaning, in this one and only instance, G-d desired that the Red Heifer be perplexing in terms of the laws of ritual impurity, so as to distance one from this species. The perplexity of these laws actually causes one to have less identification with this species, and feel alien to it. This alien feeling now combats the previous familiar feeling, and prevents one from attaching himself to this animal, for use of idolatrous purposes. The Jews are therefore atoned for their sin of the Golden Calf.