Gedanken-Experiments for Special Theory of Relativity

Let us apply the 4Ws:

What is happening? Light clock is flashing an arc . Light from the arc bounces back from the train and causes the clock to tick.
Who was watching? Train-Bob and Platform-Bob. Let us call the events $E_1$ and $E_2$ on the train reference frame, and $E'_1$ and $E'_2$ on the platform reference frame.
Where are they happening? $E_1$ and $E_2$ are occuring at the center of the train. $E'_1$ and $E'_2$ are occuring with a constant horizontal movement on every tick.
When are they happening? $E_1$ and $E_2$ are occurring cyclically; so are $E'_1$ and $E'_2$, but at a reduced frequency.

In Train-Bob's view, the light from the arc, bounces up and down vertically, causing the clock to tick. In Platform-Bob's view, the light from the arc leaves the clock at one point and returns to it some distance away. The light had to travel some distance in the horizontal direction to keep up with the train's motion.

Time between ticks (or the time intervals) appears longer to Platform-Bob. In other words, time dilates. The amount of time dilation can be easily derived using Pythagoras' theorem.

These are the quantities at play:

the vertical path of light from arc to Train-Bob as measured on the train $l$,
the time taken $t$ for light to cover above distance $l$ ($l = ct$),
the slanting path of light from arc to Train-Bob as measured from the platform $l'$,
the time taken $t'$ for light to cover the distance $l'$ ($l' = ct'$) and
the horizontal displacement of the train during the time $t'$ (given by $vt'$).

It is easy to see from the picture below that these lengths form a right triangle with $ct'$ as the hypoteneuse.

$$ \begin{array}{rl} (ct')^2 &= (vt')^2 + (ct)^2\\[6pt] \text{rearranging, }&\\ t' &= \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}\\ & = \gamma{t} \end{array} $$

In English, this states that a time interval $t$ measured by an observer on a moving frame, appears to be dilated by a factor of $\gamma$, when viewed from an observer on a resting frame.

What if we replace this imaginary light clock with real ones? Time dilation occurs for real clocks as well.

Regardless of their type, the clocks measure the time separation between events occurring at a particular location.
Regardless of the events being observed, if they occur at the same location on a moving frame, then they will be observed to occur at different locations by someone at rest.
And since the act of observation is influenced by the speed at which information about the event is relayed to the observer (i.e. $c$), the space separation also causes an additional time separation.
This space separation and the constant nature of $c$, causes the time dilation to work exactly as derived above.

Why only dilation? - because of the nature of $v/c$.

If $v < c$, then $\gamma > 1$.
If $v = c$, then $\gamma$ becomes infinity. The moving clock is frozen.
If $v > c$, then $\gamma$ becomes an imaginary number.

You could interpret the $v > c$ case in two ways:

Nothing can move faster than light.
OR if something moves faster than light, then it cannot be observed.