Mathematical puzzle

The picture has absolutely nothing at all to do with what I'm going to say - it's just a clock.

And now to work:

Here's something that's always looked like a bit of a mystery. According to Broadwcast, BBC records - specifically, a memo dated 7 July 1965 -  show that, in 1965, Television International Enterprises Ltd (TIE) 'bought [Doctor Who] prints for showing in Gibraltar, Aden, Trinidad, and Bermuda". 

Those prints were the following stories (26 episodes):

• An Unearthly Child
• The Daleks
• Inside the Spaceship
• Marco Polo
• The Keys of Marinus

And the price TIE paid was (according to Broadwcast's report of the same July 1965 memo) £1,587 2s 11d, at £75 per print.

This presents a bit of a maths problem. Whether they bought one set of prints, two sets, or thee or four sets (26, 52, 78 or 104 individual episodes), there's no way to arrive at a total price of £1,587 2s 11d if each print was £75. What you get is a range of answers from £1,950 to £7,800. 

Why is this? Well, I think the answer is actually quite straightforward, as it turns out. They weren't paying £75 per print - they were paying £75 per hour.

Gloriously, shuzbot from the Planet Mondas Forum was able to get hold of TIE's written proposal for setting up a television service in Aden (from the British Library - see this link). 

The document dates from 23 February 1962 according to the National Archives listing (there doesn't seem to be a date on the document itself, but it mentions on page three that Jamaica's TV service is due to start 'early next year', and it in fact started in August 1963).

Appendix IV outlines the costs that will be involved in obtaining programme material from the BBC and others. The bit that relates to the BBC says this:

'... The production and programme property available for Commonwealth Television Services can be obtained for Aden at approximately $40 per hour excluding freight provided that Aden Television accepts material that is being supplied to at least four other stations for which TIE (Sales) has programming responsibility' 

The document then goes on to talk about other distributors, including ITC (distributors for ATV), whose material is said to be obtainable for 'US $40 per hour under the same conditions as for the BBC'. 

Notably, it also mentions 'Granada and AR' (the latter must mean Associated Rediffusion), saying that they 'have yet to be approached over the $20 per half hour basic price' but are 'likely' to accept it. 

So: on the face of it, TIE's MO was to contact the programme distributors they proposed to buy from in advance of actually buying anything, and negotiate a flat hourly rate for all future programme purchases.

And if that equated to a cost of $40 per hour to each of five stations, as the Aden proposal suggests, then the overall rate was $200 per hour.

Was it still $200 per hour in 1965? Well, it would have been sensible for TIE to negotiate a deal that covered a number of years, so that they could have a degree of certainty about their incomings and outgoings. And, in that connection, it's worth noting that the Aden proposal itself includes forecasts for the next five years. 

The dollar to pound exchange rate didn't move around much between 1962 and 1965. In 1962, $200 was just under £71; and in 1965, it was just under £72 (there are lots of places on the Internet where you can check this). That's pretty close to the £75 figure quoted in that BBC memo.

But I think the clincher is this: I've added up the total duration of the first five Doctor Who serials, using the timings available on the Doctor Who Reference Guide site. 

Those first 26 episodes had a combined total duration of 634 minutes, 26 seconds. And £75 per hour is £1.25 per minute. 

So that would make the total cost of one set of prints (priced down to the last second), £793.04.

£793.04?! That's a bit disappointing. It still doesn't seem to fit. 

Unless ... TIE bought two sets of prints. Which, at £75 per hour, would be £1,586.08. 

And that is pretty close to the £1,587 2s 11d that they actually paid. 

Factor in the likelihood of some small discrepancies between the episode timings I've used and the ones used by the BBC (there'd need to be less than a minute's worth of difference overall), and it's close enough to persuade me that TIE did buy two sets of prints, and paid £75 per hour for them.

And that's it.