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### Thrones

Posted: **December 16th, 2008, 5:16 pm**

by **rralph**

In a magical palace there are 50 "kings", and two thrones. Each king is allowed to sit once a day on one of the thrones for 5 minutes. This happens at a random time each day and the throne is picked at random.One of the kings in the palace is cursed. If the cursed king sits on one of the thrones, then that throne also becomes cursed. Any king sitting on a cursed throne becomes similarly cursed.How many days on average would it take for all the kings to be cursed?Suppose there is a wizard in the palace who is able to remove the curse on a throne with a wave of his magic wand. However, the wizard only visits each throne once a day. How many days until all the kings are cursed?For a bonus point - What colour are the thrones?

### Thrones

Posted: **December 16th, 2008, 9:16 pm**

by **wileysw**

couple of questions to clarify the problem:each time every king randomly chooses a throne, or for every day, a throne is chosen randomly and everyone sits on this same throne?the order of kings sitting on the throne is prefixed or random?when the wizard visits, does he visit both thrones or does he randomly pick one? and i guess he does that before/after every king seats, not in between?are you into PC games?

### Thrones

Posted: **December 16th, 2008, 10:22 pm**

by **MCarreira**

QuoteOriginally posted by: wileyswcouple of questions to clarify the problem:each time every king randomly chooses a throne, or for every day, a throne is chosen randomly and everyone sits on this same throne?the order of kings sitting on the throne is prefixed or random?I chose a no lap sitting policy (the time slot for each king cannot be used by posterior kings; this is an acceptable solution for the simpler problem, without the wizard).Found 151.4 days on average (histogram on the pdf).I can modify the code to include the wizard (I'd assume that he randomly visits each throne every day, but that he is only able to lift a curse if the throne is empty, and his visits are instantaneous), but then I'd need to calculate all of the choices for each king, and the conditions become more complicated to implement.A Google search suggests that the thrones are crimson.

### Thrones

Posted: **December 16th, 2008, 11:57 pm**

by **MCarreira**

Correction: The first code did not allow for a newly cursed throne by the first king to curse the unfortunate 2nd king on the same day.Changing the code and recalculating 10,000 runs the answer is 141.7 (quite close to Sqrt[2]*2*50, maybe I'll change the parameters to see which relations hold).Corrected notebook attached.

### Thrones

Posted: **December 17th, 2008, 12:11 am**

by **mathematocrat**

151 days?? I'm obviously not understanding this problem the same way you are. Aren't we assuming that every king sits on a random throne in a random 5 minute interval each day? If that's the case, shouldn't it only take a few days to curse both thrones, and hence curse all the kings? If only 1 king sits on a throne per day, why is the 5 minute interval important? Can anyone clarify this question for dense people like myself?

### Thrones

Posted: **December 17th, 2008, 12:21 am**

by **MCarreira**

You're correct, I thought that only 2 kings would be selected each day ... back to the drawing board.

### Thrones

Posted: **December 17th, 2008, 12:29 am**

by **MCarreira**

2.757 days ... thanks for pointing it out.

### Thrones

Posted: **December 17th, 2008, 3:51 am**

by **mathematocrat**

Heh you're welcome. Three is about what I figured off the top of my head. As a side note, I'm starting to think that intuition, common sense and the ability to make simple back of the envelope calculations is all you really need in the financial world. This is why I would think physicists would be the most effective quants, since they excel at these kinds of calculations. Measure theory, stochastic calculus and other rigorous mathematical tools strike me as extreme overkill.

### Thrones

Posted: **December 17th, 2008, 11:10 am**

by **rralph**

Each king selects a throne a time at random each day and selects a throne at random every day.The wizard visits both thrones at the same time once a day (a random time each day). Assume that he always gets to visit each throne (if a throne is occupied when he visits, then he waits until the king finishes his business, then nips in and cures it before the next king sits).One King to a throne, if both thrones are occupied, then the king will have to wait his turn.The thrones could be almost any colour, but I believe one colour is much more common than any other. I will reveal all tomorrow if no-one has guessed the real motivation behind this brain teaser.One more tweak to the problem - What about if after two days of being cursed, the king turns into a frog and immediately hops out of the palace. How long now until the palace is empty (apart from the wizard)?

### Thrones

Posted: **December 17th, 2008, 1:24 pm**

by **timeds**

The thrones are white."he waits until the king finishes his business" ... was the giveaway

### Thrones

Posted: **December 17th, 2008, 2:57 pm**

by **rralph**

Correct!I thought I'd leave another clue floating around for people to get a sniff of.

### Thrones

Posted: **December 17th, 2008, 3:01 pm**

by **rralph**

I hope you all enjoyed this as much as I did.Apologies for the subterfuge, but I am quite interested in the answer. I thought that if I was a bit more cryptic about the setup then it was more likely to peak peoples interest.If anyone still cannot work out what I am talking about, let me know and I will drop a few more clues.

### Thrones

Posted: **December 18th, 2008, 1:17 pm**

by **MCarreira**

rralph, good one, although I'd probably pass on getting a sniff ...A good reminder about looking beyond the words.

### Thrones

Posted: **January 5th, 2009, 5:57 pm**

by **wileysw**

rralph, can you reveal the solution to your questions?