The Benchmark You Can Perfectly Match

A metals desk must acquire exactly $12$ lots of copper by quarter-end. The desk's execution quality will be judged against a benchmark equal to the simple average of the exchange closing prices on the quarter's final $4$ trading days. Assume you can trade any size at each close with zero fees and zero market impact, and you have no opinion on where prices are headed. How do you split up and time the buying, and why?