How Doctor Who: Thin Ice was nearly a multi-Doctor episode

As we pointed out after the episode, one of those times was with River Song, who – still sporting the Regency outfit she wore on the trip – wistfully tells Rory during 2011's A Good Man Goes To War that the Eleventh Doctor has just taken her on a special birthday outing.

“It’s my birthday. The Doctor took me ice-skating on the River Thames in 1814. The last of the great Frost Fairs. He got Stevie Wonder to sing for me under London Bridge.”

And that, of course, raises the tantalising prospect that we were potentially just a whisker away from the first face-to-face meeting between the Eleventh and Twelfth Doctors.

Sadly it didn't happen, but just how close a call was it? There are a number of factors to consider...

It’s the same year, yes, but the last Frost Fair lasted for four days before the river (Thames, not Song) thawed. And although we know that Bill and the Twelfth Doctor arrived on the final day – "last day before the thaw," the Doctor tells her after relocating the Tardis, "thought we better find a more reliable parking spot" – we don’t know exactly when River and the Eleventh Doctor were there.

At this early stage in the Twelfth Doctor and Bill’s companionship, you'd think he would try to avoid bumping into himself with a different face (Bill already asks a lot of questions). On the other hand, it’s not at all clear that he’s even capable of piloting the Tardis that accurately through time (“you don’t steer the Tardis, you reason with it… unsuccessfully most of the time”). After all, the Frost Fair was supposedly not his intended destination: “Back at the exact moment we left,” he assures Bill confidently, before stepping outside to find not a 21st-century Bristol university campus but 19th-century London.

If the Doctor can be off by over 200 years, what are the chances of him getting the right day – and if he has no control over which day they’ve arrived on, that means there’s a one in four chance that it’s the same day as River and the Eleventh Doctor.

Even then, though, they'd have to be in the right place as well as the right time. So how likely is that?

River says Stevie Wonder sang for her “under London Bridge”, so the question is, were the Twelfth Doctor and Bill anywhere near there?

When the river froze over, it generally began in earnest at Old London Bridge (demolished 18 years later in 1832), where ice broken off from the banks caught at the feet of its narrow arches, freezing the slower water.

When they arrive, the Doctor and Bill certainly step down on to the ice from a bridge, but is it the bridge? Comparing the arch we see in the episode with contemporary drawings and paintings of Old London Bridge, it’s hard to say for sure whether they’re one and the same – the real bridge had some curved arches like this one and some more angular ones. But luckily the Tardis has given us a very big clue to the Doctor and Bill’s location.

After they first land and realise where they are, the Doctor darts back inside the Tardis and examines the area on one of the view screens. It very helpfully shows us a map of the Thames indicating the position of a mysterious and very large life-form (which we later find out is the giant fuel-excreting fish which has been chained to the river bed by the evil Lord Sutcliffe).

The Doctor and Bill have materialised on the ice right above the creature's head – and if we cross-reference that point on the Tardis map with a map of the Thames from 1814 we can see that it matches exactly the location of the Old London Bridge, at Southwark.

So the bridge we see behind the Doctor and Bill almost certainly is Old London Bridge, meaning they are just feet away from where River Song and the Eleventh Doctor heard Stevie Wonder sing.

The year is right, the place is right and the timing – well, it’s certainly close, giving a one in four chance that the two Doctors were this close to crossing paths.

Of course, the Doctor told Bill that he had visited the Frost Fair, not once before, but “a few times” – and he wasn’t making it up.

In a short story that formed part of the official Doctor Who website’s 2007 advent calendar, The Tenth Doctor and companion Mai Kondon visit the very same Frost Fair in 1814. In the Big Finish audio adventure Frostfire, the First Doctor goes there with Steven and Vicki (to meet another Who favourite, Jane Austen). In series eight TV episode The Caretaker, the Twelfth Doctor promises to take Clara to the Fair. And in Doctor Who novel The Silhouette, he finally does (even if there are some possible discrepancies with the timing in that case).

So in fact, there were at least four Doctors wandering around the 1814 Frost Fair, meaning the the potential for some version of a multi-Doctor episode was pretty significant – and we could feasibly have ended up with the Tenth Doctor meeting the First, the Eleventh and two Twelves.

Now that really would have left Bill with some questions...

Doctor Who continues on BBC1 on Saturday 6th May at 7:20pm