First, some cheating solutions: (1) Connect pairs at one end. Go to the other end. Label wires and connect pairs to form a long string. Return to the first end. Each wire has some resistance, so connecting the battery and lightbulb to different lengths of the string will light the bulb to different brightnesses. Using super-human vision, you can order the brightnesses and thus determine the order of wires in the string, allowing you to label them with. This still needs two 10-mile walks. (2) Take the battery apart. Connect various numbers of cells to various pairs of wires at the first end. Go to the other end. Determine which wires are paired. Using various numbers of cells, and the lightbulb as a null detector, determine how many cells are on each pair. Only one 10-mile walk. (3) As Rich suggested, having diodes can help. (For instance, it lets you solve n = 2.) A sharp point placed near (say, the thickness of an index card) a conductive plane acts as a crude diode, assuming the battery is a kilovolt or two. Such a diode might be made from some of the patch cords. This allows various solutions, in theory, and sparks are fun. Question: Does direct current between one's tongue and hand taste different depending on the polarity? If so, that might be used in another cheat solution, but probably doesn't improve on the number of 10-mile walks. Proposal: Would this variation require the triangular numbers approach, and not the pairs-and-pairs approach?: The n wires each have a diode in them. All diodes point in the same direction, say cathode toward end A of the cable and anode toward end B. This precludes current through loops of paired wires. To make the problem solvable, suppose there is a conductive shield on the cable, which acts as one additional wire with no diode. (Or equivalently, use ground at both places. The resistance between ground connections is almost all due to the dirt resistance at the connection point, so by burying bare patch cord wire and watering the area, resistance should be low enough to use it. I know water was not one of the given tools; maybe pee a lot?) Anecdote: Several years ago, a guest speaker where I worked described his experience with communications in hostile (war) conditions. He speculated that by far the worst disruption to communications is not to cut wires, but to re-connect them in random new combinations frequently. The equipment at each end has to not only re-synchronize, but to figure out who it is now talking to, and adapt routing and high-level protocols accordingly. -- Mike