logical deduction - Labeling wires in a bundle

At a remote location, you just finished trenching a data cable across a large plot of land. The cable has 64 individual wires that are not color-coded or labeled.

You have a wire stripper, a simple electrical continuity tester, and a label maker. There is sufficient excess at either end to allow connecting and reconnecting the individual wire ends in whatever combinations you wish.

It's a long walk and you are tired. What is the fewest number of trips from one end of the cable to the other required to identify and label each individual wire in the cable?

Addenda:

• The continuity tester is a sealed unit. You can't pull the battery out and leave it behind.


• It's not necessary to make a final trip after the wires are labeled just to clean up. If they are all labeled at both ends, the job is done.

Spoiler #1

Solve the puzzle for 2 wires and extrapolate from there.

Spoiler #2

Not all spoilers are helpful.

Answer

The wires can be labelled using a total of

two trips.

We start by forming the 64 wires into 10 groups, of size 11, 10, 9, 8, 7, 6, 5, 4, 3, and 1. All the wires in each group are twisted together. We will attach a temporary label to each wire, recording the size of the group it is in.

Now, we walk to the other end. Using the continuity tester, we can determine precisely which pairs of wires are twisted together, so for each wire, we can determine the size of the group it is in, and therefore its temporary label.

Now, we will assemble the wires into 11 different groups, of varying sizes. The temporary labels of the wires in the groups will be \begin{gather*} \{11,7,6,5,4\},\{11,10\},\{11,10,9\},\{11,10,9,8\},\{11,10,9,8,7\},\{11,10,9,8,7,6\},\\\{11,10,9,8,7,6,5\},\{11,10,9,8,7,6,5,4\},\{11,10,9,8,7,6,5,4,3\},\{11,10,9,8,7,6,5,4,3,1\},\{11,10,9,8,3\}. \end{gather*} Note that no two groups are identical.

Each wire is then given a permanent label, and for each wire, we write down:

• the wire's permanent label,


• the wire's temporary label, and


• the temporary labels the wires now grouped with that wire.

We make a trip back to the other end and separate the wires. Using the continuity tester, we can see which wires are connected to which others. For each wire, we can see its temporary label, as well as the temporary labels of all other wires attached to it. This is sufficient to determine the corresponding permanent label.