Algorithm for a gear-train problem

A lathe manufacturer planned to incorporate a gear change that would produce a range of metric pitches in addition to the normal range of English pitches. The problem had previously been resolved by using a special 127-tooth change gear to compound the drive to the feedbox. However, such a large gear was awkward to accommodate and involved a loss of production time whenever a change had to be made. It was therefore decided to incorporate four smaller gears in the feedbox, making it possible to use a simple lever change to give direct drive for English pitches or back-geared drive for metric pitches. The problem was to determine the combination of four gears that would give the over-all ratio most closely approximating the one desired.