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The Emperor is annoyed that the crowd routinely chant out of step at the Empire's largest circular amphitheatre. For example, they are supposed to shout phrases in unison such as "Hail to the Emperor, he is great!" and yet they always sound like a rabble.

All powerful though he may be, he realises he can't punish the whole crowd at every event. Bread and circuses and all that.

Nevertheless he decides that just for once on his next birthday his Court Musician must arrange an event where he trains a crowd of people (the chorus) to occupy the front few rows of the arena's seats and chant a spoken birthday ode to him in perfect time.

Question

How does the Court Musician keep everyone in time?

Why does he equip everyone in the chorus with earplugs?

Where is the only place for the Emperor's throne and why?

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  • $\begingroup$ might as well add mathematics/ geometry tag? $\endgroup$ – Omega Krypton Dec 14 '18 at 10:13
  • $\begingroup$ @ Omega Krypton - I think maybe I'll add physics and geometry. $\endgroup$ – chasly from UK Dec 14 '18 at 10:34
  • $\begingroup$ +1 just for "bread and circuses". I didn't expect to learn a new (great) phrase on Puzzling. Cheers :D $\endgroup$ – BruceWayne Dec 14 '18 at 21:27
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How does the Court Musician keep everyone in time?

All need to be able to see the Court Musician who will act as a choral conductor

Why does he equip everyone in the chorus with earplugs?

So they don't go off the rhythm, misled by the other singers. Each one of them is surrounded by singers which are at ever increasing distance from them. They here a lot of people singing the same theme but slightly delayed. That's why you can see a lot of singers in pop groups covering one of their ears. This helps them hear their own voice so they don't go off tune and the conductor shows them the tempo.

Where is the only place for the Emperor's throne and why?

All singers must be equidistant so ideally the throne will be in the centre of a circle of singers. This way the sound will travel the same time the Emperor and he will hear them all singing in unison.

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  • $\begingroup$ This is all correct. Can you explain why the problem you discuss in 2, might happen? (In fact will definitely tend to happen) $\endgroup$ – chasly from UK Dec 14 '18 at 10:15
  • $\begingroup$ @rhsquared refer to my answer for the last bit $\endgroup$ – Omega Krypton Dec 14 '18 at 10:17
  • $\begingroup$ I'm sure you have this. I suppose I'm asking: If you take the Emperor out of the equation and they stand as you suggest then why would they inevitably cause problems for each other? $\endgroup$ – chasly from UK Dec 14 '18 at 10:24
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    $\begingroup$ Pop singers cover one ear to check their intonation, not to stay in time. Pop groups don't usually have the large number of players or physical separation for the timing to be a problem, and I've never seen one that follows a conductor! I'd also argue that if the crowd is indeed trained to follow the conductor, they don't really need the earplugs - this is exactly how a marching band performing a field show works. At first, it's tough to ignore the sounds coming from 50 yards away, but you pick it up after the band director/court musician yells at you enough. $\endgroup$ – Nuclear Wang Dec 14 '18 at 14:27
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    $\begingroup$ Actually there is a whole vertical line of valid positions for the emperor's throne. You're thinking 2D, but in 3D the problem looks different. (Also, the singers can be only on a semicircle, an arc, or an arbitrary subset of the circle.) $\endgroup$ – Federico Poloni Dec 14 '18 at 19:01
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Partial Answer:

Why the earplugs?

To minimize the error time when the one's voice reaches another's ear. Another type of signal, perhaps visual, is needed.

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  • $\begingroup$ What you say is correct. $\endgroup$ – chasly from UK Dec 14 '18 at 10:06
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I'll post this as an answer to my own question. Please note that @rhsquared got the whole thing right. I'm simply adding a commentary.

It is actually impossible for a full arena of people to sound to any listener in the arena as though the crowd are singing or chanting perfectly in time. Even if they are conducted by someone standing in the middle, the sound delay is always there. As @S.M. points out, even the conductor will be affected by the delay between the back seats and the front ones.

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Might be a bit of a weak answer, but

He makes them wear earplugs so that if someone makes a mistake the others won't follow, and even if there aren't any mistakes, the voice of the person next to you will come with a different delay than that of the person standing on the opposite side of the circle; puts the crowd in a circle and sits the emperor in the middle it so he won't hear any delay from the people in the back and it sounds like a "surround" sound instead, all at the same time. About keeping them on time, I don't know, guess the court musician trained them well and we can rely on the quality of his work? Besides since they're all wearing earplugs he can just visually signal them for the tempo.

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  • $\begingroup$ You're partially there. The 'science' part of the question needs to be a little more explicit. $\endgroup$ – chasly from UK Dec 14 '18 at 10:05
  • $\begingroup$ P.S. I've just realised from your answer that I missed one of the conditions out. I've added it now. Apologies. I should have thought everything through more carefully before posting. $\endgroup$ – chasly from UK Dec 14 '18 at 10:12
  • $\begingroup$ Well, I've already given the only position he can be in, which is in the center of a circle formed by the crowd, so all the sound is coming from equally distant places. $\endgroup$ – S. M. Dec 14 '18 at 10:19
  • $\begingroup$ I agree. But the reason for the earplugs is not because someone might make a mistake. Let's suppose they have all memorised everything perfectly and each practised the timing perfectly at home to a metronome. $\endgroup$ – chasly from UK Dec 14 '18 at 10:50
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    $\begingroup$ That's why I added the echo, because even if no mistakes were made there'd still be a difference in sound from the person standing next to you in the circle and the person on the opposite side of it, but I wasn't clear enough on it, I suppose. I'll edit my answer even though rhsquared's has already been accepted. Not sure "echo" would even be a correct term at all for what I meant, but it was the first thing that came to mind, my bad there. $\endgroup$ – S. M. Dec 14 '18 at 10:53

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