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# How to get a good degree 2: SHIP to DOCK

I recently received a request for help with an exercise in a text book (I guess this happens when your profile suddenly becomes higher). The exercise was a dressed up version of a children’s game. The idea is to take one word and transform it into another by changing one letter at a time. For example, to change DOG to CAT we can do the following:

DOG

COG

COT

CAT

Obviously longer words make the game more complicated but my interest in this is that it is a great way to get students talking about proof and doesn’t involve any new mathematics getting in the way of their understanding. I got the idea from Ian Stewart’s Nature’s Numbers. In Chapter 3 he talks of the necessity of proof in mathematics and to show this uses the above game played with changing SHIP to DOCK. (How this game is connected to proof is given at the bottom of this post.)

You can try this yourself. One possible answer, given by Stewart, is SHIP, SLIP, SLOP, SLOT, SOOT, LOOT, LOOK, LOCK, DOCK.

One year I set the problem of getting the SHIP to DOCK to my group of six tutees during their first tutorial in their first week of university and left them to play with it while I located some chalk. When I returned some of them had done nothing – not one letter had been changed and progress had not even been attempted.

The reason given by one of the students has stuck in my mind ever since. He shrugged in bewilderment as if I had asked him to perform brain surgery and said “But I’ve never seen this before”. He honestly could not see how he could solve such a problem without having first been given the method for it. One simple example was not enough. He was puzzled to be set a problem for which he had not already been given the key to answering it.

This student’s response is not atypical. Plenty of other students have expressed precisely this point about this and other exercises set. Basically: if you don’t tell me the method for solving a problem, then I can’t do it and in fact I will take no action to even begin.

Of course all the important problems we meet in life do not have ready made solutions. I like to tell my students that when they go out to work their boss is not going to ask them to solve a problem and say “Oh by the way, here’s how you do it”. They will have to use their brains to solve it.

The worrying aspect is that students have learned to be this helpless because of the exams they have taken over the years. None contain any “unseen” questions. This is a crucial problem we need to solve – this learned helplessness – and to be honest I don’t think I personally do enough to knock this attitude out of the students. My dream is that students would come to university wanting to tackle problems – real problems – without having already having seen the method of solution. In reality of course it is up to lecturers such as myself to attempt to inspire the right attitude. Easier said than done.

Just to finish, I should explain the mathematical point behind the exercise – it is more than just a word game. What is interesting is that in changing SHIP to DOCK Stewart says we have theorem: At some point in the chain there is a word with two vowels. The real exercise is to prove this theorem. You may like to try to solve this before continuing reading.

The key to the proof is that every four letter word should contain at least one vowel. Now to get from SHIP to DOCK the vowel has to jump from position 3 to 2 at some point. If we did this without going through a two vowel word, then we are changing two letters in one move and this is not allowed.

As an aside I think that Ian Stewart’s explanation is slightly wrong (sorry, Ian!). He uses a lemma that every English word should have a vowel. This is false as the word ‘nth’ is in the dictionary! Have a look if you don’t believe me. It’s quite funny that a mathematical word is what trips up the lemma. Thus we have to use ‘Every four letter word has a vowel’. I believe this statement is correct but am willing to be contradicted – I’m sure Scrabble fans can tell me.

http://en.wikipedia.org/wiki/Words_without_vowels

“Hymn” is exactly 4 letters long. “Rhythms” is the longest in common usage.

Except that y is effectively a vowel in this case.

“psst” has no vowel sounds.

http://en.wikipedia.org/wiki/List_of_words_in_English_without_vowels

Incidentally, this is is a (mildly) disguised problem in graph theory. The set of n letter English words is the vertex set. There exists an edge between two vertices A,B if changing 1 letter in word A gives word B. (This relation is symmetric so there’s no need to resort to a digraph.)

The problem is then of finding a walk from word A to word B. Any graph searching algorithm such as breadth first search or depth first search will do quite nicely.

Of course, noting that the vowel changes speeds up the process because we then have an intermediate location C and we want a walk A to C and C to B.

I may look for a decent dictionary file in a bit and find some properties of this graph.

Thanks James!

Psst is good! If we allow abbreviations (which we don’t, of course!) then we can probably find a chain from SHIP to DOCK that doesn’t have two vowels by using asst (for assistant) and so on.

I am utterly astonished by your students’ reactions to this simple task! (I did it in six by the way – SHIP, SNIP, SNAP, SOAP, SOAK, SOCK, DOCK. I think I can see why a word with two vowels is necessary at some point, but will need to think more carefully than is possible at quarter to two in the morning!). I have to be honest and admit that if I’d been in your class I’d have been wondering ‘what on earth’s this lecturer up to?’ but would have got on with it nonetheless. Anyway, I do seem to remember word games like this when I was at junior school.

Thanks for the comment. The student’s reaction is surprising. It is perhaps not surprising that he left during his second year.

As for thinking “what on earth’s this lecturer up to?”, well that’s the sort of reaction I want. It makes the reveal of the reason more memorable!