Friday Puzzle -- Unlabeled cable

20 miles

Method

There is a general method that works for any number of wires. This method takes advantage of the fact that the number Alex chose (120) is a 'Triangular Number':

Divide the wire into 15 bundles of 15,14,13,12,11,10,9,8,7,6,5,4,3,2,1 wires each. Twist the ends together and label 'A'-'0'.

Drive 10 miles to the other end and use the battery & bulb to ring out the bundles and label them (also 'A'-'O').

Take one wire out of the 15 in bundle 'A', Ground it and label it 'Aa'.

Connect the next wire from 'A' to an unconnected wire from bundle 'B' and label the two wires 'Ab' and 'Ba'.

Continue connecting each wire from 'A' to an unconnected wire from a different bundle and label them similarly ('Ab'/'Ba' through 'Ao'/'Oa').

Follow the same procedure with each bundle of wire until all wires are connected and labeled. At this point, each wire should be connected to exactly one other wire from a different bundle or ground and should have a two-letter unique label.

Drive 10 miles back, untwist the original bundles (but keeping track of which wires were with which bundle), ring out and label that end.

The labels will be:

Aa Ab Ac Ad Ae Af Ag Ah Ai Aj Ak Al Am An Ao

Ba Bb Bc Bd Be Bf Bg Bh Bi Bj Bk Bl Bm Bn

Ca Cb Cc Cd Ce Cf Cg Ch Ci Cj Ck Cl Cm

Da Db Dc Dd De Df Dg Dh Di Dj Dk Dl

Ea Eb Ec Ed Ee Ef Eg Eh Ei Ej Ek

Fa Fb Fc Fd Fe Ff Fg Fh Fi Fj

Ga Gb Gc Gd Ge Gf Gg Gh Gi

Ha Hb Hc Hd He Hf Hg Hh

Ia Ib Ic Id Ie If Ig

Ja Jb Jc Jd Je Jf

Ka Kb Kc Kd Ke

La Lb Lc Ld

Ma Mb Mc

Na Nb

Oa

Re: Method

Alex mentioned two solutions, but I can't think of another with as few miles.

Re: Method

There is one other 20 mile solution I can think of. While it saves a lot of driving, you'd spend a lot of time ringing out wires at both ends.

Re: Method

Well, let's see it.

If it is the one I know, it makes use of two things not used in the first method, namely the ground and the fact that the power source and indicator (battery and bulb) of your continuity tester are separable. And it leads to numbering the wires 1-120 directly.

An alternative labeling scheme.

Exactly right.

No difference in substance, but an alternative, after labeling the members of groups A-O on each end, is to take one from from each group, twist them together, and label them all with "1" to go along with their letter; one from each remaining letter group, labeled with "2" in addition to their letter code.. , At the other end, untwist all the letter groups and find the number groups, adding appropriate number labels to each wire.

As I said, no difference in solution, but an alternative way of expressing it may help some see it.

Another method

Twist wires together in groups of 2. You will have 60 pairs.

Drive 10 miles to the other end.

Pick one wire, label it '1' and connect it to ground.

Use the battery & bulb to find the wire it is connected to. and label it '2'.

Pick another wire,label it '3' and connect it to wire '2'.

Find it's partner and label it '4'. Continue until all wires are connected and labeled.

At this point you have a long continuous path to ground and each wire is labeled with its distance to ground.

Now drive 10 miles back to the start.

Untwist one pair. One of the pair will have path to ground, one will be open.

Count how many wires (remember, each pair is 2 wires) still have continuity to ground. Label your grounded wire with this number. Label the open wire with that number plus one.

Re-twist the wires and repeat for all other pairs.

Very good!

And yours?

What about the method that separates the battery and bulb?

Essentially the same

After returning to the original end, disconnect all the pairs electrically, but leave them paired physically. Then look for the hot wire, labeling it "1" and its partner "2", then twist them together and look for the next hot wire to label "3", etc.

I think that is a _slightly_ shorter solution than yours. At each step, this solution would have you test on average about 1/2 the unconnected wires to find the next hot one, while you have to test all the unconnected wires to find what number to assign.

Turns out you do not need separable indicator and power source. I had not thought that through clearly. Here is the solution I saw elsewhere:

"First, tie the 120 wires together randomly in 60 pairs. Next, go to the far end, randomly label any wire 1, and connect the battery to it. Test which other wire is tied to it at the starting end, and label that wire 2. Then pick another wire other than 1 or 2, label it 3, and tie it to 2, so now the battery is connected to 1, which is tied to 2 at the other end, which is tied to 3 at the end you're at. Now test which wire is tied to 3 at the other end, and label that 4, etc. What you will wind up with is all 120 wires tied to each other in a continuous sequence. Then go back to the end you started at, leaving the battery behind, connected to wire 1. Before you untie all the wires at the starting point, label each wire so that you know which wire was paired with which. Now with all the wires untied at the starting point, test which wire is connected to the battery, and label that 1. Whichever wire was in the same pair as 1, label that 2, and then tie 1 and 2 back together. Now you can find 3, because it's tied to 2 on the far end. Once you find 3, label the wire it was tied to 4, etc."

I had a problem with that, based on my physics 101-level understanding of electricity. What are the other sides of the battery and lamp connected to? That is why I specified the grounding sheath, but as you pointed out, that lets you use a simple Radio Shack continuity tester, not a separate bulb and battery. Maybe the original solver was connecting to local ground at each end? I don't know if that would work.

Re: Essentially the same

That is considerably faster than my solution.

Thanks for another great puzzle!