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I went ahead and tried a number of possible continuations of this sequence, leaving out different numbers.
The Lagrange polynomial utility was never able to predict any one of the missing numbers, any which way.
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I input 1,2,4 for x, and 1,12,1 for y. This means our original series with the 3rd number missing: 1,12,...,1 
The calculator provided P(x) = 12 which translates to their answer being 1, 12, 12, 1 --- but it should be 1,12,1,1 
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Next, I input 1,2,5 for x, and 1,12,1 for y. This would be for our series with the 3rd and 4th numbers missing: 1,12,...,...,1
The calc gave P(3) = 15.67 and P(4) = 12 which means their answer is 1, 12, 15.67, 12, 1 --- should be 1,12,1,1,1
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Next, I input 1,2,6 for x, and 1,12,2 for y. This is for our series with the 3rd, 4th and 5th missing: 1,12,...,...,...,2
The calc gave P(3) = 17.60, P(4) = 17.80, P(5) = 12.60 or, 1, 12, 17.60, 17.80, 12.60, 2 --- should be 1,12,1,1,1,2
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When I left out the first number things got strange.
2,3,5 for x, 12,1,1 for y outputs 30.33, 12, 1, -2.67, 12 --- instead of 1,12,1,1,1,2
2,3,4,6 for x, 12,1,1,2 for y outputs 42, 12, 1, 1, 2, 4 
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This formula is utterly useless on this puzzle. The formula clearly has zero relation to the puzzle key. 
Sometimes relying on pure mathematics puts you everywhere EXCEPT in the right place.
This polynomial calculator is entirely unable to anticipate any one of the numbers in the subject series! 

Neil Obstat,

what about deaf persons who can't hear the chime? They would have to watch the cuckoo. Don't you remember what you said about the advantages of hourglass sand timers?

Westminster chime for example has 4 different melodies it plays at the 4 quarter hours.
There are a variety of other chimes but the Westminster is the most popular of the musical chimes, in the western world, anyway.

With new European Union copyright laws the U.K. will have to pay to continue using Georg Friedrich Händel's original work used by the clock tower at the Palace of Westminster (Westminster Quarters). I assume that this will break their economy.


Since you're not a friend of cuckoos, here an alternative from Augsburg:

With new European Union copyright laws the U.K. will have to pay to continue using Georg Friedrich Händel's original work used by the clock tower at the Palace of Westminster (Westminster Quarters). I assume that this will break their economy.

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How many ways can you get stuff wrong, Struthio? Handel originally composed the symphony the Westminster Chime was BASED on, but he did not compose the chime. In fact, it is only the four pitches that occur in his symphony at bars 4 and 5 that are used in the Chime, which has 5 variations that comprise its composition. The Westminster Chime was composed by either Regis Professor of music, Dr. John Randall, or his pupil, William Crotch in 1793. The carillon was first played in the bell tower of St. Mary's the Great, the University Church, in Cambridge. Later it was reproduced on a large scale in the Victoria Tower Clock of the House of Parliament (renamed Elizabeth Tower in 2012).
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Speaking of which, here is another puzzle!
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What does how many tones in the Westminster Chime separate the midpoint of the only repetition of the same two pitches from the beginning of the repetition of the entire sequence of tones have in common with the variable pitches themselves?
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And another one:
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If the name of the structure that houses the Great Clock of the House of Parliament, is Victoria Tower, why is it popularly referred to as "Big Ben?"
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And another:
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If the Westminster Chime is rendered in the key of D, what pitch and octave does Big Ben have?
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Neil Obstat,

can you prove that the puzzle in the OP

Can you move just one digit to a new location, making this equation true?
Do not move signs, only move one numeral.

82 - 43 = 1  

beside the single solution, found by MaterDominici:

82 - 3⁴ = 1    

has no other solutions?

Are you aware of BREXIT?

Sure. That's why I expect retaliation by the European Union.

what sparked Brexit: the Brits were not about to pay duties.

Sitting in a pub, not being about to pay duties, I would stay for another beer, not leave.

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In general, the thing that makes good puzzles so charming is that they have one unique solution that is much simpler than all the runner-up answers. It is the beauty of simplicity that sets them apart. The simpler, the better!
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If the name of the structure that contains the Great Clock of the House of Parliament is "Clock Tower" (or "Victoria Tower" renamed Elizabeth Tower in 2012), then why is it popularly referred to as "Big Ben?"
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A simple question with a simple answer.
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The movement and technical description of the various parts of this historic clock are a topic of vast interest by clockwork enthusiasts worldwide. Some of the numbers and materials describing its parts and operation are frankly hard to believe, when considered out of context. The fact that it's been in service under all weather conditions for 159 years and continues to operate reliably is quite a testament to its designers and craftsmen who made it a reality. It survived the onslaught of German bombs in WWII, with the peals of clock chimes not missing a beat for the duration even amidst the explosion of ordnance. There have been lists of trivia facts drawn up about this "Great Clock" (a moniker in use for over a century already) that are quite amusing -- especially once you know a thing or two about pendulum clocks. Yes, it is regulated by a pendulum, with a period of two seconds; 13 feet tall, weighing 660 pounds, and costing a lot more! Incredible, in an age of electronic everything. It is 100% mechanical, a purist's dream come true. It gets wound three times a week by hand, a process that takes 90 minutes. For the English (thank goodness) it would be UNTHINKABLE to replace this enormous movement with a more modern one that would work automatically, perhaps by wind power. Some have said that they were a little short-sighted by not making the actual pendulum VISIBLE through a window of some kind on the sides of the tower. The regular 2-second beat of this clock's heart could be made to appear as a sparkling Jєωel in the center of the tower. In answer to this charge, it has been countered with the fact that this is a FOUR-SIDED CLOCK and its pendulum operates in only two directions, so to make it visible, only two of the four sides would be able to display its movement. Well, that's not a closed issue. By use of prisms or mirrors, the image of the pendulum in motion could conceivably be conducted to appear similarly through windows in each of the four sides. The cost of making such changes do not qualify as urgent, however, when compared to paying for essential maintenance. Certainly, four TV screens displaying the PICTURE of the pendulum in motion could be hung on each of the 4 sides, but that would not be in keeping with the longstanding character of the clock: all mechanical, no electricity required. The whole ambiance of the Great Clock is a cultural icon of the British people. In 2007 it was found to be the single most popular tourist attraction and thought to be the single most culturally relevant item on English soil. IOW Big Ben is as British as motherhood and apple pie is American. They've just managed to turn their national symbol into a tourist attraction, unlike Americans. But having their "pie" and eating it too comes with a price! As of 2017, necessary maintenance and functional renovations are in progress which are estimated to cost £61 million. During this repair process the chimes will be silent for several months. We should hope this will not stir up civil unrest.