I'm interested in what happens in those 'fuzzy' periods when solving mathematical problems, when you are not sure what to do. The only way I can see of getting to grips with this is to work on mathematical problems myself, and see what this tells me about the problem solving process.

To this end, I decided to choose a problem at random from Thinking Mathematically and make an audio recording of what I was doing verbally (

**see bottom of page*). I worked on the 'Jacobean Locks' problem (p. 176):

A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to. How many locks are required, and how many keys?

Here's what happened. What does it tell me (us?) about how people solve mathematical problems?

*[After reading the question a few times, trying to understand the situation...]*

"Not much information... How many locks? How many keys? Imagine there was say three locks... and three people. Three people could have one each...

"If five people in the village, and three locks A, B, C, then each person could have three keys each {a, b, c}... Is that it? ... No, no, that wouldn't work at all, because any

*one*person could open the locks!"

"So it might be something like {a,b}, {a,c} and {b,c}... But then any

*two*people could open the locks... It's more tricky than I thought... I've got to start afresh... but not sure how to...

*[I try imagining the situation, spend some time thinking about six people (thinking I might extend the solution with three people) with various numbers of locks, and various cyclic arrangements of keys, but with no success, and then go back to a simpler version]*

"Let's say there's only three people on the island, with three locks... with keys {a}, {b} and {c}, that's fine! ... So with

*four*people with three locks, one with key {a}, one with {b} and one with {c}... then there's a problem because the fourth person could not have any one of these, or any two of them, so it's not possible... So with four people, you'd have to have four locks, is that correct? But how would you arrange them? They have {a,b}, {b,c} and {c,d}.... but the fourth one... it doesn't seem possible...

*[I look at this for a while, but just can't see how to make the cyclic arrangement work for four people four locks, so I try a different number of locks. I try various cyclic arrangements for 6 locks - considering the idea that the number of locks might be something to do with permutations... I have a vague sense that this problem is like a Latin square, or a block design, hence the idea of cycles...]*

"Let's go back to four people... My question again is: I've got 4 people, how many locks do I need, and how do I distribute the keys?

*[There are longer gaps between talking now. I'mI finding it difficult to talk and give my full attention to the problem.*

*I'm trying various arrangements on paper,*

*forming ideas about what arrangement do and don't work, what is and isn't possible.]*

"The bottom person, the fourth person, has to complete the set for any three, but not for any two, the gaps each person has, if I leave the gaps separate then it can't work...

"Six locks? Four locks? I don't know how many locks... three people three locks, four people...? Perhaps I could think about giving people the keys first, and then finding the number of locks to make it work...

"I still think my idea of cycling is a good one. So... maybe we could have like {a,b}, {a,c}, {a,d}, the fourth person would need...? Maybe {e}, a fifth one? Repeating {a, ...} doesn't feel right.

"Perhaps I need a different way of representing it. What about what each person

*can't*have? For three people and three locks, having two keys missing for each person works, but for four people and four locks, having three keys missing doesn't work...

"The problem I've got is with four people, I don't know how many locks I need... Is it four people four locks? The first person has {a,b}, the second person has {a,c}, the third person could have {b,d}, I'm just trying something different... But then the second and third person could open it, so I can't have that...so the third person could have {b,c}... So now any two people I pick have keys {a,b,c}... well, the fourth person comes along and has {d}... That works!

*[I think this is the solution and go to bed, but then suddenly realise that it doesn't work - if the first three people try to open the locks! I'm in bed at this point, and have stopped recording the audio. I start to sketch some diagrams in a notebook...]*

I start using these rectangle grids (in which rows = people and columns = locks):

*each column must have exactly two gaps*! This

*must*be the case if any three people can unlock any particular lock. Then all I need is to find an arrangement in which no two rows have all of the letters, but can't *quite* make it work here.

I realise I need something more systematic; I arrange the gaps systematically like this, but it is still does not

*quite*satisfy the constraint that no two people can open the chest :

** I am aware that what is going on for me is not only that which I can verbalise. I am certain that there are also things are happening below the level of my conscious awareness, which when coming to conscious awareness feel like what we might call 'insight'.*