Given that $x$, $y$, and $z$ are the amounts invested by $X$, $Y$, and $Z$ respectively, and their investments are in the ratio $x : y : z$. The profits are in the ratio $z : y : x$.

The profits are proportional to the product of the amount invested and the time period of investment. Let's denote the periods of investment by $T_X$, $T_Y$, and $T_Z$.

We have the equation:

$x \times T_X : y \times T_Y : z \times T_Z = z : y : x$

This implies:

$\frac{x \times T_X}{z} = \frac{y \times T_Y}{y} = \frac{z \times T_Z}{x}$

Simplifying, we get:

$\frac{T_X}{\frac{z}{x}} = \frac{T_Y}{1} = \frac{T_Z}{\frac{x}{z}}$

To find the ratio of the periods $T_X : T_Y : T_Z$, let's rewrite this in a common form:

$T_X : T_Y : T_Z = \frac{z}{x} : 1 : \frac{x}{z}$

Now, if we multiply through by $x \times z$ to eliminate the fractions, we get:

$T_X : T_Y : T_Z = z^2 : x \times z : x^2$

Thus, the correct answer is:

$\boxed{z^2 : xz : x^2}$

This corresponds to option (c) z$^2$ : xz : x$^2$.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would the ratio of periods change if the profit ratio was instead $x : y : z$?
2. If the investment amounts are $2$, $3$, and $4$, what are the periods $T_X$, $T_Y$, and $T_Z$ for the corresponding profit ratio?
3. Can the method used here be applied if the investments were functions of time?
4. How would the answer change if the investment ratios were squared or cubed instead?
5. What if the profits were in the ratio $x^2 : y^2 : z^2$? How would that affect the time period ratio?

Tip: When analyzing problems involving ratios and proportions, it’s crucial to identify what quantities are being compared and how they are interrelated before simplifying the ratios.