To determine the length of one year on Mercury, we use Kepler's Third Law, which states that the square of the orbital period $ T $ is proportional to the cube of the mean radius $ R $ of the orbit:

$ T^2 \propto R^3 $

Given:

The radius of Mars' orbit around the Sun, $ R' = 4R $, where $ R $ is the radius of Mercury's orbit.

The Martian year $ T' = 687 $ Earth days.

First, apply Kepler's Third Law to find the ratio of the orbital periods:

$ \left(\frac{T'}{T}\right)^2 = \left(\frac{R'}{R}\right)^3 = \left(\frac{4R}{R}\right)^3 = 4^3 = 64 $

From this, we can solve for the ratio of the periods:

$ \frac{T'}{T} = 8 $

This means that Mars takes 8 times longer to orbit the Sun compared to Mercury. Therefore, the length of one year on Mercury is:

$ T = \frac{T'}{8} = \frac{687}{8} \approx 85.88 \text{ days} $

Hence, one year on Mercury is approximately 86 Earth days.