The Antikythera mechanism is an ancient computing device of astonishing complexity and sophistication. Crafted by engineers between 150 and 100 BC, using (what appear to be) Greek theories of astronomy and mathematics, it is the oldest known scientific calculator and has even been called an analog computer. It was recovered in the year 1900 and was found to contain over 30 gears. It was up to scientists, historians, and engineers to reverse engineer the purpose of the device, a feat no doubt comparable to its original construction. 
It is often the case that a device such as the Antikythera cannot be observed directly to figure out how it works. Science is also this way, in that we can observe what happens, but may not know why it happens.
In this exercise, the challenge is to guess the equations for the unknown functions labeled , , ..., by evaluating these functions at various input values and observing the corresponding function values.