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In the 50th anniversary special, when Clara told him to be a Doctor, he thinks up a plan:

DOCTOR: Gentlemen, I have had four hundred years to think about this. I've changed my mind.

WARRIOR: There's still a billion billion Daleks up there, attacking.

DOCTOR: Yeah, there is. There is.

DOCTOR 10: But there's something those billion billion Daleks don't know.

DOCTOR: Because if they did, they'd probably send for reinforcements.

CLARA: What? What don't they know?

Then they realize something in their minds that surprises them:

DOCTOR: This time, there's three of us.

WARRIOR: Oh! Oh, yes, that is good. That is brilliant!

DOCTOR 10: Oh, oh, oh, I'm getting that too! That is brilliant!

DOCTOR: Ha, ha, ha! I've been thinking about it for centuries.

WARRIOR: She didn't just show me any old future, she showed me exactly the future I needed to see.

MOMENT: Now you're getting it.

DOCTOR: Eh? Who did?

WARRIOR: Oh, Bad Wolf girl, I could kiss you.

MOMENT: Yeah, that's going to happen.

DOCTOR 10: Sorry, did you just say Bad Wolf?

CLARA: So what are we doing? What's the plan?

WARRIOR: The Dalek fleets are surrounding Gallifrey, firing on it constantly.

DOCTOR 10: The Sky Trench is holding, but what if the whole planet just disappeared?

CLARA: Tiny bit of an ask.

DOCTOR 10: The Daleks would be firing on each other. They'd destroy themselves in their own crossfire.

WARRIOR: Gallifrey would be gone, the Daleks would be destroyed, and it would look to the rest of the universe as if they'd annihilated each other.

CLARA: But where would Gallifrey be?

DOCTOR 10: Frozen. Frozen in an instant of time, safe and hidden away.

DOCTOR: Exactly.

WARRIOR: Like a painting.

Then on Gallifrey the 3 Doctors tell the Council the plan:

DOCTOR: We're flying our three Tardises into your lower atmosphere.

DOCTOR 10: We're positioned at equidistant intervals around the globe. Equidistant. So grown up

WARRIOR: We're just about ready to do it.

GENERAL [OC]: Ready to do what?

DOCTOR: We're going to freeze Gallifrey

GENERAL: I'm sorry, what?

DOCTOR 10: Using our Tardises, we're going to freeze Gallifrey in a single moment in time.

WARRIOR [on monitor]: You know, like those stasis cubes? A single moment in time, held in a parallel pocket universe.

DOCTOR: Except we're going to do it to a whole planet.

DOCTOR 10: And all the people on it.

GENERAL: What? Even if that were possible

GENERAL [OC]: Which it isn't, why would you do such a thing?

DOCTOR: Because the alternative is burning.

DOCTOR 10: And I've seen that.

DOCTOR: And I never want to see it again.

GENERAL: We'd be lost in another universe, frozen in a single moment. We'd have nothing.

DOCTOR: You would have hope. And right now, that is exactly what you don't have

GENERAL: It's delusional. The calculations alone would take hundreds of years.

DOCTOR: Oh, hundreds and hundreds.

DOCTOR 10: But don't worry, I started a very long time ago.

Then other TARDISes come into the sky, which turn out to be the Doctor's past incarnations:

DOCTOR 1: Calling the War Council of Gallifrey. This is the Doctor.

DOCTOR: You might say I've been doing this all my lives.

DOCTOR 2 [on monitor]: Good luck.

DOCTOR 3 [on monitor]: Standing by.

DOCTOR 4 [on monitor]: Ready.

DOCTOR 8 [on monitor]: Commencing calculations.

DOCTOR 5 [on monitor]: Soon be there.

DOCTOR 7 [on monitor]: Across the boundaries that divide one universe from another.

DOCTOR 6 [on monitor]: Just got to lock on to his coordinates.

DOCTOR 9: And for my next trick.

GENERAL: I didn't know when I was well off. All twelve of them!

ANDROGAR: No, sir. All thirteen!

(the 12th Doctor Peter Capaldi comes in)

ANDROGAR: Sir! The Daleks know that something is happening. They're increasing their fire power.

GENERAL: Do it, Doctor. Just do it

GENERAL [OC]: Just do it.

I'm confused: what comes in their mind when 11 says "this time there's 3 of us"? What calculation did he start a very long time ago? Why was the Doctor's best strategy for saving Gallifrey to get help from his 13 incarnations?

12

what comes in their mind when 11 says "this time there's 3 of us"?

The fact that there are three of them.

After centuries of bemoaning what he was forced to do, he's hit by the realisation that he isn't forced to do it any more, because with three Doctors now available at the same time, there are also three TARDISs, and thus the ability to implement a "freeze Gallifrey" plan that it seems would have been impossible to achieve on his own.

What calculation did he start a very long time ago?

The answer is mostly there in the dialogue; the rest can be deduced by reason.

  • The Doctor(s) planned to freeze Gallifrey in a pocket universe;
  • The calculations to achieve this would take hundreds of years with Time Lord technology;
  • The Doctor has been alive for hundreds of years;
  • Recruiting his first incarnation and telling him to start the calculations, means they'd be done by the time Doctor 11 was up (a trick they'd already used earlier in the same episode, to introduce it to us).

Why was the Doctor's best strategy for saving Gallifrey to get help from his 13 incarnations?

  • The TARDISs needed to be positioned equidistant around the globe, suggesting that position and coverage is important while surrounding the planet to freeze it;
  • Since we've just made it so that all the Doctor's prior incarnations are aware of the proceedings, why not get them all to pop along and add extra power? Extra equidistant planet coverage? We already established that the plan would work with three TARDISs but not only one, so quantity and/or physical distribution in space is clearly a factor.
  • Who else would you get to do it? And why?

Out-of-universe obviously it was supposed to be a "cool" moment, but in-universe I think a much more interesting question is why no future Doctors beyond 13 were present too.

  • I might add that the "starting a calculation in the past and having it complete in the present" was previously used in the episode, in the Tower of London "escape"; this is presumably why the Eleventh Doctor's line "this time, there's three of us" was a lightbulb moment for the other two – Jason Baker Apr 9 '16 at 22:07
  • i thought the what comes in their mind when 11 says "this time there's 3 of us"? mean they start seeing their 13 incarnations William Hartnell, Patrick Troughton, Jon Pertwee, Tom Baker, Peter Davison, Colin Baker, Sylvester McCoy, Paul McGann, John Hurt, Christopher Eccleston, David Tennant, Matt Smith in there head and realises all of their incarnations are the only pepole to help them – Andrew Casali Apr 9 '16 at 22:08
  • @AndrewCasali: I think the thought that comes in their mind is precisely what they say it is. – Lightness Races in Orbit Apr 9 '16 at 22:19
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    I was typing up an answer too, but this nails it perfectly; +1. @AndrewCasali If you want a one-line summary, try this: if one Doctor is good, then three is even better and thirteen is brilliant. All those lives a single Doctor has saved, all those planets he's rescued - with thirteen times the amount of awesomeness, he can even accomplish the feat of saving Gallifrey. – Rand al'Thor Apr 9 '16 at 22:22
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    in-universe I think a much more interesting question is why no future Doctors beyond 13 were present too - obviously, it's because they haven't ever had existed yet, but one day they will have been. :-) – Harry Johnston Apr 9 '16 at 22:58
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Sorry to resurrect such an old question, but my take on the "I'm getting it now too" is that at some point ten or eleven manage to communicate with one, and from that point onwards in some pseudo-time the later incarnations become aware of it.

Naturally this means they've changed their personal history again...

It's a bit like the dénouement of Bill and Ted's Excellent Adventure, really.

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    While I am not a fan of DW, so I don't know if this answer is correct, but there is nothing wrong with answering old questions. There are even badges for it! – Skooba Dec 15 '17 at 14:22
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Everything in the Day of Doctor is in Duality as in it happens in twos.

1 No more
2 Two Versions of the 7th Doctor on the screen
3 2 Doctors, 2 Queens, 2 Lethbrigs Stewarts.
4 The black archive 2 pictures of Susan on the Board plus 2 Companion's in the same pic.
5 Both Doctors wear glasses and try the fez on.
6 The thing that detects the Zygon the doctor says it goes ding once then he then says it goes ding twice.
7 The Glass smashes inwards then outwards Etc.

Watch it again you will see its all in twos.

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