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Did you see my edit? Do we have to use all the coins? Or can you simply pick two coins, if both heads you're done (two piles of 0 tails up each), both tails you're done (two piles of 1 tails up each), and in the case of 1 heads 1 tails simply flip one of them and you're done?
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The two piles - together - should contains all the coins in the room...
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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I think I figured it out. My last guess (though it didn't account for all the coins) was on the right track. Take a simple example of TTTHHHHHHH (7 heads, 3 tails). If you take three coins you'll get either {TTT}, {TTH}, {THH}, or {HHH} leaving {HHHHHHH}, {THHHHHH}, {TTHHHHH}, or {TTTHHHH} respectively. Those don't match up, but if you notice, flipping what was taken yields equal tails (example: {TTH} -> {HHT}, {THHHHHHH} = 1T each).
This flipping works by removing coins equal to the number of tails. By subsequently flipping the coins, this set now contains a number of a heads equal to the number of tails removed from the original set and a number of tails equal to the number of heads removed from the original set. This naturally balances out the number of tails in each set.
An example: 10 tails {TTTTTTTTTT}. Let's remove 6 tails {TTTT}. Those 6 tails are now in the taken set {TTTTTT}. We need 10 total coins though, so we must have taken 4 heads as well {TTTTTTHHHH}. Because we removed the same number of total coins as the number of total tails, whatever amount left in the original set in tails must be represented as heads in the taken set. Thus after the flip we now have {HHHHHHTTTT} which has 4 Ts just like the original set {TTTT}.
So for your question: Pick 14 coins, any coins, and flip them. Done.
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Right answer. Over-complicated explanation... 
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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But the only answer that explains why it works 
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Let see...
After you separate 14 coins from the rest there is exactly n coins with tails in that pile...
In the other pile there is 14 - n...
Now flip the coins in the pile of 14, now it has 14 - n coins with tails, just as in the other pile...
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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Your explanation doesn't account for why taking an amount of coins equal to the number of tails is necessary  But yes, the rest of the explanation could be simplified by math instead of words 
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Gather 14 coins at a time. Feel them if found any tails keep them separate. Repeat until you find all 14 tails and then gather rest of them ( All heads ) and make two piles with same number of coins from head and tail pile. Best case you find all 14 in first go. Worst case you have to feel all 284 coins.
Zen and the art of software maintenance : rm -rf *
Maths is like love : a simple idea but it can get complicated.
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I can live with the best case, the worst case bothers me however... There is much simpler way to do it 
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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Turn the lights on ?
Zen and the art of software maintenance : rm -rf *
Maths is like love : a simple idea but it can get complicated.
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I'm not sure there is a switch inside the room, but can 'feel' for it...
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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Turn 14 coins over in one pile and all the others go into the second pile.
Whichever 14 you turn over, there will be the same number of tails on both piles.
veni bibi saltavi
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Nice and clean!
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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That works, but my answers were more entertaining
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Well spotted.
I wanna be a eunuchs developer! Pass me a bread knife!
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Tell them where the couch is. They will all go and hide. No coins, no problems.
"It is easy to decipher extraterrestrial signals after deciphering Javascript and VB6 themselves.", ISanti[ ^]
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Easy.
Just say "Thank God we don't live in Medieaval times", and turn the light on.
I wanna be a eunuchs developer! Pass me a bread knife!
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If we do not live in medieval times, why should we thank God?
And al-Khwārizmī - who lived even before - was able to solve it without electricity...
Skipper: We'll fix it.
Alex: Fix it? How you gonna fix this?
Skipper: Grit, spit and a whole lotta duct tape.
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Make a pile of all the coins.
Take 14 from there to make the second pile.
Turn all the coins in the second pile upside down.
the pile: H:270 T:14
two piles: H:270-t T:t; H:t T:14-t (t is whatever number of tails in the big pile)
upside down the small pile: H:270-t T:t; H:14-t T:t (Done).
Ooh, ooh.
(Anyone notice the error in my answer to the ant on the rope? - should have been exp(100)-1 seconds. I had stopped the ant at the starting point of the other post rather than where it had moved to. - So I now agree with the other responder except I think he used 364 days in a year??)
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Member 12255252 wrote: Anyone notice the error in my answer to the ant on the rope?
Yes, we all noticed. That's why we don't believe anything you say, any more.
I wanna be a eunuchs developer! Pass me a bread knife!
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I'm sure nothing can go wrong from us playing God. /sarcasm
Jeremy Falcon
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God was once approached by a scientist who said, “Listen God, we’ve decided we don’t need you anymore. These days we can clone people, transplant organs and do all sorts of things that used to be considered miraculous.”
God replied, “Don’t need me huh? How about we put your theory to the test. Why don’t we have a competition to see who can make a human being, say, a male human being.”
The scientist agrees, so God declares they should do it like he did in the good old days when he created Adam.
“Fine” says the scientist as he bends down to scoop up a handful of dirt.”
“Whoa!” says God, shaking his head in disapproval. “Not so fast. You get your own dirt.”
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Pualee wrote: You get your own dirt.
Jeremy Falcon
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