The more I think about it, without numerous calculations as you have done, the more I have realised that I was hoping to be lucky with the first cut in order to reduce the number of cuts and waste of cable.
Exactly - as I said at the start, it involves the concept of 'utility' - i.e. whether you are 'risk-averse', 'risk-neutral' or 'risk-loving'.
As you say, the non-central first cut involves gambling that you will be lucky with that first cut, in which case you will obviously 'win'. However, you are obviously more likely to be unlucky than lucky - so,as we've seen, that means that 'on average' there will be no gain and, probably a small 'loss'. The person who takes that approach is therefore towards the 'risk-loving' end of the utility spectrum.
It's exactly the same with traditional gambling. Most people actually understand that if one takes part in purely random gambling (roulette, lotteries etc.), 'on average' (i.e. 'in the long term'), one cannot win. However, the 'risk-loving' person will concentrate on the fact that they could win a fortune with one of their earliest roulette stakes or lottery tickets.
Having said that I still believe that starting in the middle and halving is not the wisest plan as you will never gain an advantage due to luck. The odds are evens that you could guess which side of the centre cut the fault may be but even if correct there is no advantage.
Again, a 'risk-loving' view. Yes, you will sometimes gain an advantage by not starting in the middle - but, more commonly, you will 'gain' a disadvantage and, as I have illustrated, those disadvantages slightly outweigh the advantages so that, on balance, they are somewhat worse off. 'Risk-loving' (as in compulsive gamblers) is largely about psychology - the 'buzz' that a person gets by occasional wins, even if they are more than cancelled by other losses.
As an example, take a Ring with sixteen sockets - one of which has a break in a conductor. Dividing the circuit in the centre and halving each time will result in having to make four divisions every time before finding the faulty socket. Whereas quartering, If you happen to choose the correct side still results in having to make three divisions. If you chose the wrong side and start halving it could be as many as five. Should you continue to quarter it could take seven - worst case scenario. It would seem that it is a gamble.
Indeed - and, coincidentally, very similar to the figures from my 100m cable simulation. However, everything I've said above still applies.
Obviously, what 'on average' means in this context is this ... Assuming locations of faults are totally at random (i.e.absolutely no other information), take two electricians and get them both to undertake one of these exercises every day for a few years - one using 'halving' and the other using 'quartering'. The 'quartering' one would feel chuffed, because (s)he will have sometimes located the fault with only 3 divisions, something that the 'halving' one will never have done. However,when they totted up, they ought to find (assuming the installations they'd been testing were, on average, 'comparable') that, over the years, the 'quartering' one had undertaken more divisions/cuts than the 'halving one'. So, it's a toss-up between the occasional 'short term' win (and feeling of satisfaction.'buzz') and a long-term (probably fairly small loss).
It may be worth taking a guess on the first cut hoping to be lucky but if not halving appears the best policy after all.
If it's a pure (random) guess, then, even if it 'feels good', there is not going to be a (long-term' or 'on average') benefit. However, if there is any 'information' one can feed into the exercise (e.g. practicalities as to where the neighbour could have got at a cable to damage it), selecting one's first cut/division on the basis of such additional information can obviously be very worthwhile.
Kind Regards, John.
Whilst, in terms of computing, it's not necessarily the most efficient search algorithm (particularly because of its requirement for sorting - or, at least, variants using some sort of indexing), there really aren't (to the best of my knowledge) any better approaches available to the electrician with a faulty length of cable or ring final circuit to fault-find.
