Let's tackle this step by step:

A$)$ To determine the number of possible routes and enumerate them, we need to identify all the routes from City A to City C via City B$.$ Since all routes pass through City B$,$ there are essentially three segments to consider: from A to B$,$ from B to C$,$ and from A to C directly. Here are the routes:

$1.$ Route $1$: A $->$ B $->$ C

Links utilized: $1,2$

$2.$ Route $2$: A $->$ C

Links utilized: $3$

$3.$ Route $3$: A $->$ B $->$ C

Links utilized: $4,2$

So$,$ there are three possible routes for this journey.

B$)$ Now, let's find the traffic flow patterns and equilibrium travel time for each link and route at equilibrium:

I. Traffic Flow Patterns:

To find the traffic flow patterns, we need to apply the principles of User Equilibrium $($UE$),$ which states that each traveler chooses the route with the minimum travel time. Since there are $150$ individuals seeking to travel from A to C$,$ we need to distribute them among the available routes in a way that minimizes their travel time.

Let's denote the traffic flow on each route as $\backslash ($ Q$\_$i $\backslash ),$ where $\backslash ($ i $\backslash )$ represents each route. At equilibrium, the total travel time for each route must be equal.

Using the given cost functions, we can set up equations for each route and solve them simultaneously to find the traffic flow on each route.

For Route $1$:

$\backslash [$ t$\_1($Q$\_1)=0\mathrm{.}2$x$\_1+3\backslash ]$

$\backslash [$ t$\_2($Q$\_1)=$ x$\_2+6\backslash ]$

For Route $2$:

$\backslash [$ t$\_3($Q$\_2)=0\mathrm{.}1$x$\_3+1\backslash ]$

For Route $3$:

$\backslash [$ t$\_4($Q$\_3)=0\mathrm{.}3$x$\_4+2\backslash ]$

$\backslash [$ t$\_2($Q$\_3)=$ x$\_2+6\backslash ]$

II$.$ Equilibrium Travel Time:

Once we find the traffic flow on each route, we can calculate the equilibrium travel time for each link and route.

For each link, the equilibrium traffic flow $(\backslash ($ x$\_$i $\backslash ))$ is the sum of traffic flows on all routes that use that link.

The equilibrium travel time for each link is then calculated using the given cost function with the equilibrium traffic flow value.

Let's calculate these values step by step.