This essay was originally published on my (now-defunct) Substack newsletter on 09/11/23.

“I don’t understand this Five Doctors scene. Plz explain?” – Mr. Laser Beam (www.trekbbs.com)

I’ve been meaning to write this post for around three years. Back in 2020, I was kindly invited onto the Galactic Yo-Yo podcast and the very first thing Molly and I discussed was this scene about from The Five Doctors. I’ll admit I was actually struggling a bit1 because at that point in time I’d only seen the entire story once, and it had been a few years ago. I’ve watched the episode a couple more times since then. But yes, I did recall watching this and thinking ‘well, that doesn’t make any sense.’ It was, frankly, an entirely futile conversation, but we gave it a good go. Now I’m going to write an entirely futile blog post, and I’m going to give it a really good go.

“Surely you know some basic mathematics, child?” – The First Doctor, The Five Doctors

Let’s start with the concept of pi. What is pi?

Well, it’s the sixteenth letter of the Greek alphabet and the equivalent to ‘p’ in the English alphabet. But it also has a mathematical definition. Pi is, as Tegan correctly defines it within the episode, “the ratio of the circumference of a circle to its diameter.” This holds true for any circle, no matter how big or small; that’s why it’s so special.

It’s also an irrational number, meaning it cannot be expressed as a fraction, and so it has a never-ending decimal expansion. If I asked you to think of a combination of digits, any combination of any length, it’s somewhere within the decimal expansion of pi. It might take a very long time to find it, possibly even more time than you have left to live, but it will be there. The First Doctor also correctly quotes the decimal expansion to eight decimal places – “Three point one four one five nine two six five” – during the scene. I can do it to ten decimal places from memory so I like to think that makes me cleverer than Dr Who.

So anyway, that’s the theory. But as the First Doctor also says, “Oh, but the application. The application.” The application is where all the confusion happens.

“It’s a pun, stupid.” – HandeToon (www.youtube.com)

Having watched the scene several times over, I cannot infer any consistent logic or visual grammar that explains how the number pi is relevant to which characters survive crossing the 10×10 red-and-white checkerboard and which characters don’t. The script does not specifically dictate where each character should step and it seems fairly obvious that neither the actors nor the director have discussed how they should actually do this. On the all-new commentary track for The Five Doctors – 40^th Anniversary Edition on the recently released Doctor Who: The Collection – Season 20 Blu-ray boxset, Peter Davison tells moderator Matthew Sweet that this was “precisely the detail [director Peter Moffatt] was not on top of.”

It is particularly egregious for the Master, who we see crossing the board forwards and backwards more than once, and he takes a different path every time! He even steps off the board at one point before reaching the end! If you listen to the ‘secret’ DVD commentary track on the original transmission version of The Five Doctors you can hear past and present Doctor Who producer Phil Collinson saying “It’s such nonsense,” whilst past and present Doctor Who actor David Tennant can be heard saying that they are “Walking over whatever squares they like.”

There are probably still some Doctor Who fans watching this scene who are thinking “I’m not very good at maths, that’s why I don’t understand this bit.” Don’t worry, it’s not you – it’s the episode. There are no complicated mathematical ideas actually in play here; it’s just a bit of silly wordplay.

“I’m guessing they just didn’t care about the solution making sense.” – u/SillyNonsense (www.reddit.com)

I could happily sit here and come up with a number of ways why the number pi might have been applied to this scene, but that would just be purely speculative. This is exactly what I did during the podcast. What I’m more interested in here is what the writer Terrance Dicks actually intended for this scene, if anything. Did anyone ever actually bother to ask him this? Well, he does talk about this on the DVD/Blu-Ray commentary track for The Five Doctors – Special Edition. He says “It doesn’t actually make sense… I’m innumerate, I can’t do mathematics.”

Thanks Terrance. Good of you to come clean.

And frankly, it’s obvious. The First Doctor refers to pi as a “mathematical formula”. It isn’t. It’s just a number. A formula would be an equation for working something out, like using the Pythagorean theorem to work out the side lengths of right-angled triangles. You remember Pythagoras (c.570 BCE – c.490 BCE) from school, right? Weird fella, had his own cult. Anyway, the point here is that Terrance is trying to come up with a clever-sounding solution to the problem he has created here and he’s decided to use maths as a short-hand for ‘clever’. How flattering, if unoriginal.

Then Tegan ends the scene with the quippy reply “I hope you’ve got your sums right.” Wait, so now there’s more than one calculation happening here? What are you going on about, Tegan? Maybe she just wants to join in with the mathematical wordplay. It’s what Adric would have wanted.

‘Never mind what it’s written in, what does it say?’ – The Brigadier, The Five Doctors novelisation, p93

Hang on – I’ve just had a thought. What about the novelisation? Terrance Dicks wrote a novelisation of The Five Doctors; it somewhat infamously arrived on bookshop shelves before the UK transmission date. How does he explain it there?

The Doctor frowned, studying the board.  ‘A hundred squares, ten by ten… So, using the first hundred terms of pi as co-ordinates – Yes, that’s it, it must be.  Let me see now, three point one, four…’  The Doctor began mumbling a long stream of figures, faster and faster. At last he stopped. ‘Yes, that’ll be it!’

Tegan never did understand quite how the ‘safe’ sequence worked, even when the Doctor (her Doctor) explained it to her later. All she could gather was that if you could observe exactly where the lightning bolts struck each time, and if you could then carry out some terrifyingly complex mathematical calculation at blinding speed, you might then be able to work out a way of crossing the second part of the board without setting off the trap.2

Sounds clever, doesn’t it?

But actually, I think this extract helps illuminate some of the aforementioned dialogue from the TV version of the story. The First Doctor’s mention of a ‘mathematical formula’ fits more neatly if it is being used to calculate the safe sequence of squares across the board. It can also help explain why each journey across the board is different if the safe squares keep changing. The position of the next safe square can be identified using the next number(s) in the sequence calculated from the formula – a formula that is based around the digits of pi, whatever that might look like! Even Tegan’s pun finally has some context, as there’s more than one sum being performed now!

So you know what? I think Terrance has been a touch modest here. There’s actually some level of thought that’s gone into this. It’s just that the absence of any technical explanation in the script and the absence of any technical direction in the performances leaves the whole thing looking like a load of nonsense on the television.

Terrance Dicks is a wily writer and script editor; he’s very good at managing the limitations of his knowledge and his memory. It’s why, for example, he avoids giving specific years in the stories he script-edited for during the Pertwee era, just to give himself that wiggle room for plausible deniability. He knows he will never be as clever as the Doctor, or even the collective audience watching at home, because he knows some of them are actual mathematicians like me. But he can settle for the perspective of a confused Tegan. We get ‘pi’, and ‘co-ordinates’, and ‘figures’, and ‘calculations’ that are ‘blinding’ and ‘terrifying complex’. Nothing really substantial, but it all sounds so clever. And only a fraction of that cleverness has made it into the final scene as filmed.

I’ll leave the last word to Sue who really manages to… sum all this up.

“That scene was a disaster.” – Sue Perryman (www.wifeinspace.com)

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Footnotes

[1] Even listening back to it now, I can still hear the sound of my brain trying to remember what actually happened in the scene.

[2] Doctor Who: The Five Doctors novelisation by Terrance Dicks, p82.