To determine the intercepts on the crystal axes by the plane (231) in a simple cubic lattice, we need to analyze the Miller indices of the plane and their corresponding intercepts.

A simple cubic lattice consists of equally spaced lattice points in three mutually perpendicular directions, represented by the crystal axes.

The Miller indices for a plane in a crystal lattice are determined by taking the reciprocals of the intercepts made by the plane on the crystal axes and reducing them to the smallest possible integers.

To find the Miller indices for the plane (231), we need to determine the intercepts on the crystal axes.

Step 1: Determine the intercepts
The intercepts on the crystal axes are determined by the reciprocals of the coefficients of the Miller indices. In this case, the Miller indices are (2, 3, 1), so the reciprocals are (1/2, 1/3, 1/1).

To find the intercepts, we take the reciprocal of each fraction:
x-intercept = 1/(1/2) = 2
y-intercept = 1/(1/3) = 3
z-intercept = 1/(1/1) = 1

Step 2: Simplify the intercepts
The intercepts obtained in Step 1 are in their simplest form.

Step 3: Determine the ratio of intercepts
The ratio of intercepts is calculated by dividing each intercept by the smallest intercept obtained in Step 2. In this case, the smallest intercept is 1.

x-intercept ratio = 2/1 = 2
y-intercept ratio = 3/1 = 3
z-intercept ratio = 1/1 = 1

Therefore, the ratio of intercepts on the crystal axes by the plane (231) in a simple cubic lattice is 2:3:1, which corresponds to option B, 3:2:6.

Hence, option B is the correct answer.