So Graeme, what’s the secret?

Now, it’s not often I get mistaken for a sphinx, and since the question came out of the blue I was admittedly a little unprepared.

My interlocutor was a lecturer in plumbing and gas at West College Scotland (Paisley) and he was remarking on the recruitment of nearly thirty adult trainees in the area compared to only four plumbing apprentices.

Probably a lucky alignment of stars, I offered, hoping I could turn the conversation onto Capaldi’s first appearance as Dr Who. No such luck.

So we discussed economic conditions, employer confidence, previous experiences with apprentices or trainees, our relationship with the industry, and to be honest I wasn’t so much articulating a list of well thought through reasons as producing an Emsworthian dribble of stating the bloody obvious.

Either way we were both dissatisfied by these well trodden explanations because much of the macro economic condition of the construction industry experienced by electrical contractors was bound to be similarly experienced by plumbing contractors.

There was also, I suggested, a cultural norm of apprenticeship recruitment within the electrical contracting community such that even at the depth of the economic recession the industry was still recruiting four hundred plus apprentices and adult trainees.

That was a kind of interesting thing to hear myself say, and I was left wondering how such a cultural norm is produced or established.

Alongside doing a Future Learn MOOC, ‘Basic Science: Understanding Numbers‘, I’ve started reading Daniel Tammet’s book *Thinking in Numbers*. The first chapter, ‘Family Values’, uses set theory to explain how his nine siblings were considered legion in his local community. It’s a grand read. This way of thinking was new to me, and I’ve spent the last forty-eight hours in a bit of a maths induced daydream. I think I’ve grasped the basics and so here it might be necessary to take a deep breath.

In space an adult trainee is either in place x, or some other place, y. It doesn’t matter where exactly, the trainee is either here or there. Now the same applies for every other trainee, individually, and combined in every possible configuration with any of the other trainee or trainees in the set.

So, if we start with the plumbing apprentices and looking at them as a set, this can be written as

S = {p1, p2, p3, p4}

The set of all subsets is calculated by multiplying 2 by itself by the toal number of objects in the set. So for the plumbing apprentices the set of all subsets is calculated by multiplying two, by itself, four times. This is equal to

2 x 2 x 2 x 2=16

which means that there are sixteen different ways to spot one or more of these apprentices. I would suggest that given the large area of Renfrewshire and Inverclyde that they’ve been recruited in, bumping into a plumbing apprentice in the area seems unlikely.

However, if we do the same with the group of twelve adult trainee electricians that started last week, the set can be written as

S = {at1, at2, at3, at4, Aat5, at6, at7, at8, at9, at10, at11, at12}

with the subset of all sets being equal to multiplying two, by itself, twelve times. This is equal to

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 4,096

So now, and with only three times as many members in this set, there are four thousand and ninety-six different ways to spot one or more of these trainees: this I find incredible, and it goes a long way to explain how a cultural norm is constructed. Even though there are only twelve of them it must seem like these guys are everywhere.

**References**

File:Louvre sphinx.jpg, Wikimedia Commons, http://upload.wikimedia.org/wikipedia/commons/5/51/Louvre_sphinx.jpg, accessed 25th August 2014

Es un día fantástico con sol. (Dëdalus), taken by Ignacio Sanz, Flickr, uploaded on 23 August 2008, https://www.flickr.com/photos/ignaciosanz/2798127903/in/photolist-5gg8Wx-72sWKe-8SW8pp-8SWgq4-9megxg-72sWDT-72wWsd-njNt7m-95X8a4-73X6jk-72wWxL-74vt4h-84r8BZ, accessed 24 August 2014, CC-BY-SA 2.0

Tammett, D, (2012) *Thinking in Numbers: How Maths Illuminates Our Lives*, Hodder & Stoughton, London