Mostly Harmless

The best of the Two Tribes problems - Part 2

Continued from here.

You point to one of the paths and say:

"If I were to ask you if this path leads to the village would you say pish?"

This solution follows the normal pattern for this type of problem - but the pish/tush confusion is a novel twist. In our hut in Devon none of us (I and my brothers) succeeded in arriving at this solution but I do remember working out why this still works even when you don't know whether pish means yes or no (Mr Gardner left that part as a teaser).

I remember sitting at a trestle table and making a little grid out of knives and forks. I labelled the columns and rows with other domestic items and then filled in the grid with the answers given by the native. Salt meant pish and pepper meant tush. The resulting pattern was as follows:

	Path goes to village	Path does NOT go to village
Pish means YES
Pish means NO

From this I could see that if the native answers pish then the road goes to the village - regardless of whether pish means yes or no. In the general case, if the native uses the same word in the answer that you used in the question then the path leads to the village.

I remember the dawning sense of triumph I got from from examining this boolean decoration. Mum was less impressed: "That doesn't quite count as setting the table..."

Posted at 12:45PM Apr 27, 2007 by John Alderson in Problem Solving  |  Comments[0]

The best of the Two Tribes problems - Part 1

We were on holiday in a hut in a remote spot in Devon when I was eleven or twelve years old. On sunny days we walked down to the beach (about two miles away) and played all day, returning at night with as much driftwood as we could carry. On rainy days we burnt the driftwood on the hut's hearth which was backed by a fat black pipe thrown into folds like the body of a prone python. The water heated in the pipe rose by convection to a half-buried tank some way up the hill behind the hut.

Most of a rainy day was spent indoors reading books from the hut's ample stock of paperbacks and it may have been there that we discovered the first collection of Mathematical Games made from Martin Gardner's long-running column in Scientific American and published by Pelican - or we might have brought it with us. At any rate, it was there that we read his round-up of "Two Tribes" problems, the pinnacle of which went something like this:

You are an anthropologist visiting a remote island inhabited by two tribes. Members of one tribe always tell the truth, members of the other always lie. Apart from this detail they are indistinguishable from each other and happily coexist in a prosperous village some distance inland.

To reduce disturbance from visiting anthropologists the natives have built a fork in the one road leading inland from the shore. One path of the fork leads to the village where travelers can be assured of safety and hospitality; the other leads to a dangerous swamp where they will be instantly devoured by voracious giant land-snails.

However, the natives are not unsporting and there is always one of their number stationed at the fork to provide assistance. By ancient tradition you are permitted to ask him or her ONE question which will be answered yes or no. Unfortunately you have lost your Remote Islandese dictionary in the rough sea crossing and, though you have a working knowledge of the language, you cannot remember the words for "yes" and "no". That is to say, you know the words are "pish" and "tush" but you can't remember which is which.

Your problem is to construct one question that you can ask the native at the fork in the road, the answer to which will guarantee that you know which path leads to the comfort and safety of the village and avoids the ravages of the snails.

(As before, I have disabled comments on this entry to prevent spoilers.)

(PS: I may have made up the bit about the snails...)

Posted at 05:42PM Apr 24, 2007 by John Alderson in Problem Solving  | 

SGRT meets The Reluctant Surgeon

As promised, I attempt to shine the beneficent light of an analytical troubleshooting method into the dark recesses of the pesky Reluctant Surgeon problem.

The method in question is Sun Global Resolution Troubleshooting which is Sun Microsystems' implementation of Kepner-Tregoe's Resolve product. I will not give an exhaustive account but just show ways in which conscientious application of the method might uncover the concealed sticking point in the problem.

I was prompted to do this by the following question:

SGRT reminds you to list your assumptions but how can I list an assumption of which I am not conscious?

I can think of three ways in which SGRT might offer the key of this problem to a trouble shooter. Approaches 1 and 2 use the following problem statement.

Problem Statement

SGRT encourages us to frame a problem statement in terms of a divergence from what "should" pertain. The "should" here is that the surgeon should operate. The surgeon is not operating and we need to find out why.

First Key - Trivial

SGRT generates a set of questions which are used to establish precise details of what the problem IS and what it could be but IS NOT. (More about this later.) The answers to these questions are then used to ask more speculative questions to determine differences between the IS and IS NOT cases.

Having established initially that this particular surgeon won't operate but another one might, SGRT would lead us to ask a question like What is different or unusual about this reluctant surgeon?" to the person posing the problem. An honest answer in the UK would have to supply the information that the surgeon is a woman because, in the UK, only 6% of consultant surgeons are - so it is unusual. If the "IS NOT" surgeon was a male (likely) then the sex of the reluctant surgeon would count as a difference. Of course, in a real problem situation the person reporting the problem may ignore differences that they consider irrelevant, but this is why SGRT encourages us to "Question to the void" saying (repeatedly) What else is different or unusual about this surgeon? until even the most banal answers have been given.

However, this method is inadmissible for our problem because we are not allowed to ask questions. We can only suggest solutions. This situation is not uncommon in the real world. A customer might not be able to interrupt his production to provide more data, or we may be foolishly working on a problem while key people are unavailable (i.e. burning the candle at both ends).

So we have to suggest our own answers, and this too offers a key.

Second Key - Requires devotion

The first of the "IS" questions establishes in what object the problem resides. Answer "The surgeon" because "The surgeon IS reluctant". We then move to the "IS NOT" side of the same question:

What other surgeon could be reluctant but IS NOT?

Here is where we need to be conscientious. It is easy to fill in the blank here and simply answer "Some other surgeon". If this were an IT problem with an SFW series 1000+8i then this would be like answering "some other computer" or "my laptop". The idea is to find something as similar as possible to the problem object which nevertheless does not exhibit the problem. e.g: "Another SFW series 1000+8i in the same room" or "An SFW series 1000+4i".

In our case this makes us analyse the idea of kinship. We may think the surgeon is a deluded man. Perhaps he knows the boy slightly and is lying because he doesn't want to operate. Perhaps his use of the word "Son" is aspirational. We must put all that aside and simply list the surgeons who legitimately could and could not operate. If we are conscientious our table will have a "cannot operate" column including the word "mother".

Third Key - Shift of focus

It's irritating. We are not allowed to ask questions but the room is full of grinning idiots who know the answer and tell us it's easy. So that suggests a new problem statement.

I cannot understand the Reluctant Surgeon Problem

If we repeat the analysis with this problem statement then we will certainly start analysing our assumptions because the problem is now seen to be inside our own head. Because we have redefined the problem statement we can now start asking and answering questions about our mental model of the problem. We might be lucky and trip over our fatal assumption while listing all the assumptions we know we have made.

For instance, we might look at what elements are missing from our model (since it is we who are missing something); the other driver in the crash, the other medical staff, the other relatives (presumably telephoned) of the boy...

This list will also throw up the key insight that this is not a problem with a surgeon but a problem with inconsistency. It is a "someone must be lying" problem when we know they are not. The IT equivalent might be a silent data corruption problem where the entire datapath is protected by ECC logic. The hidden assumption is that data is being corrupted somewhere (because that's what it looked like when the problem statement was drawn up); the actual cause is that the application at one end of the path was built with later header files than the application at the other end. Addressing the inconsistency is key.

In the surgeon problem, focusing on the inconsistency causes us to give equal weight to the man driving the car as to the surgeon. After all - the fact that he is the boy's father is equally as problematic as the fact that the surgeon claims to be. "Someone must be lying".

If we ask the question "Who else might have driven the boy to the match but did not?" we are even more likely to think of the mother than if we ask it of the surgeon.

---

All in all the conclusion seems to be that an analytical troubleshooting method is only as good as the degree to which you follow it conscientiously. It is not just about filling in a template but about evaluating what you know and seeing what parts of it are wanting, and looking critically at whether what you know really justifies the current problem statement.

Posted at 12:36AM Dec 09, 2006 by John Alderson in Problem Solving  |  Comments[0]

Mother! That's it!

I commend to you this small addition to the world's bank of colloquialisms.

When I was at school some evil friend had laid upon my little circle the curse of the Reluctant Surgeon Problem. All of us had attained enlightenment (after various embarrassing intervals) save one poor unfortunate, who was still wrestling with it two days later. I think he was working on some scheme involving identical twins and amnaesia.

Anyway, in the middle of a rather subdued physics lesson he suddenly cried out "Mother! That's it!" to general laughter and the consternation of the physics teacher. This phrase then became popular with us for any situation where a long awaited insight finally arrives - something like "eureka!". This is a modest attempt to revive it.

I admit it sounds a little outré, but you will have the quiet satisfaction of knowing what it really means.

Posted at 06:56PM Oct 26, 2006 by John Alderson in Problem Solving  |  Comments[0]

Mysteries of Cognition - part 2

So, what is so interesting about the Reluctant Surgeon Problem? What always piqued my curiosity was not that a few people take days to figure it out but rather that no one (ok, almost no one) answers it instantly (me included). The most common exchange goes something like:

Ah! The surgeon is the boy's stepfather!
No.
Ok. The surgeon is the real dad but the guy killed in the crash is the boy's adoptive father, right?
No..
So the boy has two biological fathers, using IVF!
No... (sigh)

Even if the fourth suggested answer is the correct one this little paddle through the first three takes some explaining. Somehow the brain is battling an unconscous assumption.

The usual explanation is simply one of cultural stereotype, surgery having been traditionally a male dominated profession. I think this is only the beginning. If you think in pictures it may be that as the boy is wheeled up to the operating table your mind has provisionally supplied a male surgeon standing ready with his instruments, even if you yourself are a woman. However, I think the popularity of medical documentaries and soap-operas featuring many high-powered female consultant surgeons these days makes this a less than complete explanation.

The language of the problem may be a second part of the explanation. There are no feminine pronouns. The boy is a boy. He and his father are off to a football match - which conjures more masculine imagery (OK -that's cultural stereotype too...) By the time we reach the operating table the problem-solving part of our mind is primed with fathers and sons. This may take the form of a flawed syllogism:

All fathers have sons
The surgeon has a son
Therefore the surgeon is a father

I think the cultural stereotype is still key. I assume that a mother driving her daughter to ballet would far more frequently elicit the instant answer the problem deserves because it does not work at cross purposes with the stereotype. But what about a father driving his daughter to ballet? I reckon substituing a daughter for a son would also help weaken the hold of the unconscious assumption.

This is a testable prediction as long as too many people don't read this blog (there doesn't seem to be any danger of it...)

The third part of the explanation is simply that the problem is presented as a problem at all. As soon as we are told something is a problem we are put on guard for impending mental strain. What's more, we are told it is a lateral thinking problem. It is not. It is a vertical thinking problem. Ask a computer with only limited grasp of natural language and family relationships and it will spit the answer out instantly. But knowing that we have to identify a "problem" forces us to make the surgeon a man - even though this thought is never consciously articulated.

To me this surgeon story (since it is not a problem at all) is the archetype of a very real and pernicious enemy to the person who solves problems for a living - the "unconscious assumption". In problem solving we are often asked to "list our assumptions", but it is hard to list assumptions of which we have no awareness. So I am going to return to this surgeon thing again and look at how a formal problem solving method might fare at cracking it - and to see if such a method needs enhancing to tickle unconscious assumptions.

More of a bleeding lecture than a blog you might say, but then you can always go and watch an episode of ER or Holby City (ah, Connie...) instead...

Posted at 07:53PM Oct 25, 2006 by John Alderson in Problem Solving  |  Comments[2]

Mysteries of Cognition - part 1

The following problem (sometimes, oddly, classed as a lateral thinking problem) is as old as the hills, but in spite of this there never seems to be a shortage of people who have never heard it, but many of these do have the stamina to persevere with it for hours until they have satisfactorily figured it out. I will say more about it in a later posting, but for now here it is:

A father and his son drive off to see a football match. After an enjoyable game they are driving home in the dark when they are involved in a horrific traffic accident. The father is killed outright and the son is rushed to hospital with internal injuries. But in the operating theatre the surgeon, on seeing him, exclaims "I cannot operate on this child, he is my son!"

How can this be so?

Posted at 02:58AM Oct 24, 2006 by John Alderson in Problem Solving  |