The Quiet Route: Solution

Weekly Puzzle #05 of the Autumn 2026 season presented The Quiet Route, a courier dispatch map wrapped in a Wayne Enterprises Applied Sciences work order. With Weekly Puzzle #06 now active, ranked submissions for Issue 05 have concluded.

Spoiler warning: This post contains the complete solution for Weekly Puzzle #05. If you have not reviewed the transmission yet, you can inspect the archived puzzle record.

1. The Challenge Artifact

The transmission asked for a discreet delivery: send a courier from START to END with the smallest total delay, then submit the letters stamped by the links it actually traverses.

WAYNE ENTERPRISES / APPLIED SCIENCES
Dispatch approved: L. Fox
Vehicle return: before the board meeting.

All links are one-way. Delays add.
A courier begins WITHOUT a permit.
Only arrival at B issues a permit.
Entry to D requires that permit.
An issued permit lasts for the journey.

from   to    delay stamp
START  A     2     B
START  C     1     X
A      B     3     E
A      D     1     Y
B      C     1     A
B      END   15    Z
C      D     2     C
C      END   14    W
D      E     1     O
E      END   3     N

The table describes a small directed, weighted graph. Each link has a delay (its edge weight) and a stamp (a single letter). The answer is not the delay itself, but the sequence of stamps collected along the optimal legal route.

2. The Trap: Ignoring the Gate

If the permit rules are ignored, this is a textbook shortest-path problem. Two routes tie for the minimum delay:

Both are invalid. Each reaches D without ever visiting B, so the courier arrives at the gate without a permit. The tie itself is a useful warning sign: a well-formed puzzle with a single accepted answer is unlikely to hinge on two equally cheap routes.

3. Location Alone Is Not Enough

The rules give the courier a piece of memory: whether a permit has been collected. Two arrivals at the same node are not interchangeable if one carries a permit and the other does not.

This matters because a standard Dijkstra search that tracks only the node fails here, even when it checks the gate:

1. C is first reached via START → C at delay 1, without a permit. The search marks C as settled.
2. Later, START → A → B → C reaches C at delay 6, with a permit. Because C already has a cheaper label, this arrival is discarded.
3. From the settled, permit-less C, the link to D is blocked, so the search falls back to C → END and reports XW at delay 15.

The cheaper arrival at C does not dominate the more expensive one: it cannot continue to D, while the expensive one can. The fix is to search over an augmented state space where each state is a pair:

$$\text{state} = (\text{location}, \text{permit})$$

Here permit is either no or yes. This doubles the number of states but restores the property that Dijkstra relies on: once a state is settled, no later path to that same state can do better.

The transition rules follow directly from the artifact:

• Following a link adds its delay and appends its stamp.
• Arriving at B sets permit = yes; nothing ever clears it.
• A link into D is available only from states where permit = yes.

4. Enumerating the Legal Routes

The graph is small enough to list every legal route from START to END:

The two cheapest-looking shortcuts into D (START → C → D and START → A → D) are blocked, so every route that uses the fast D → E → END tail must first detour through B. The detour costs A → B → C = 4 more delay than START → C alone, but it unlocks the gate and still beats every alternative.

The 12-unit route is unique; the next-best legal route costs 15.

5. Verifying with Dijkstra

The state-augmented search fits in a few lines of Python:

import heapq

links = [
    ("START", "A", 2, "B"),
    ("START", "C", 1, "X"),
    ("A", "B", 3, "E"),
    ("A", "D", 1, "Y"),
    ("B", "C", 1, "A"),
    ("B", "END", 15, "Z"),
    ("C", "D", 2, "C"),
    ("C", "END", 14, "W"),
    ("D", "E", 1, "O"),
    ("E", "END", 3, "N"),
]


def quiet_route():
    # Each search state is (location, has_permit), not just location.
    queue = [(0, "START", False, "")]
    settled = set()
    while queue:
        cost, node, permit, stamps = heapq.heappop(queue)
        if node == "END":
            return cost, stamps
        if (node, permit) in settled:
            continue
        settled.add((node, permit))
        for start, end, delay, stamp in links:
            if start != node or (end == "D" and not permit):
                continue
            heapq.heappush(queue, (cost + delay, end, permit or end == "B", stamps + stamp))
    return None


print(quiet_route())  # (12, 'BEACON')

Following the optimal route and reading each link's stamp in order:

The stamps spell:

BEACON

Submitting BEACON completed the issue. Letter case and repeated whitespace are ignored by the answer checker.

6. Constrained Routing in Practice

The mechanics of The Quiet Route reflect a family of problems that appears throughout computer science and operations research:

State-Space Augmentation

Whenever the legality or cost of a move depends on history, the graph's nodes are no longer sufficient as search states. The standard technique is to build a product graph whose nodes combine a location with the relevant history, such as a collected key, remaining fuel, or elapsed time window. Game pathfinding uses the same idea for key-and-door levels, and model checkers explore products of program locations and variable values.

Resource-Constrained Shortest Paths

Vehicle routing and logistics systems frequently require that a pickup precede a drop-off, that a driver hold a certification before entering a zone, or that a route respect time windows. These are variants of the resource-constrained shortest path problem. A single boolean permit keeps the state space tiny; richer resources lead to label-setting algorithms that track and prune non-dominated labels at each node.

Policy Routing in Networks

Network routing faces similar constraints. Interdomain routing policies, MPLS traffic engineering, and firewall traversal rules can make a numerically shorter path unusable. The cheapest path by metric is not always a path the network is allowed to take.

Wayne Enterprises Theming

The work order's framing—Wayne Enterprises, Applied Sciences, and a dispatch approved by "L. Fox"—refers to Lucius Fox and his quiet support of Batman. The board-meeting note is office cover-story flavor; it adds no deadline or routing rule. As with the Portal storage test in Issue 04 and the Matrix operator console in Issue 03, the theme provides narrative texture, while the challenge itself rests on standard graph algorithms.

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