You are a sage in service to a king of a profoundly troubled kingdom.
A sordid spectacle...
Every ten years, the king holds a grand tournament, inviting the 1,000 most suitable princes to compete for the hand of one of his daughters. Regrettably, the king's true motive lies in exploiting the princes' legendary oneupmanship.
At the beginning of the tourney, the king has his steward present the sitting princess with a gift (a piece of jewelry) in the presence of all 1,000 princes so that they can witness her delight. He then invites them to introduce themselves one at a time, and watches contentedly as a sordid spectacle of oneupmanship unfolds.
The first prince to approach, wanting to vastly outdo the steward, pledges to give the princess one piece of jewelry for every day of the tourney.
The second prince, wishing to vastly outdo the first, pledges that every day of the tourney he shall give the princess as many pieces of jewelry as the first prince gives her that day plus as many pieces of jewelry as the first prince has given her on all previous days.
Likewise, the third prince, determined to vastly outdo the second, pledges that every day of the tourney he shall give the princess as many pieces of jewelry as the second prince gives her that day plus as many pieces of jewelry as the second prince has given her on all previous days.
This mad spectacle of devotion continues for all 1,000 princes, ending when the 1,000th prince has boldly pledged that every day of the tourney he shall give the princess as many pieces of jewelry as the 999th prince gives her that day plus as many pieces of jewelry as the 999th prince has given her on all previous days.
The plot thickens...
The grand irony of the king's scheme is that he hates jewelry. He allows the princess to collect it only so that he can give it to the nobles attending his decennial Nobles Gala, which he holds after the tourney is completed.
There are 24 nobles he must invite to the gala, and they're notoriously fickle. For one thing, those who attend demand some jewelry in return for their attendance. For another, the attendees insist on each receiving exactly the same number of jewelry pieces as each other.
Worst of all, rumours abound that some of the invitees may not show up for the upcoming Nobles Gala. Perhaps all 24 will, perhaps only one will, or perhaps any number in between. The exact number is unknowable until the gala is held.
A sage is needed!
As the king's sage, issues related to distribution of jewelry naturally fall to you. He makes it clear to you that:
- regardless of how many nobles ultimately turn up at the gala, their fickle demands must be met
- the only jewelry that will be distributed among the attendees is the jewelry given to the princess during the tourney
- every single piece of jewelry must be given to the attendees. Throwing it away (which is a curse) or keeping it (which the king hates) is unacceptable.
The only thing in your power to control is the number of days the tourney will last. It must last at least one day (obviously), and ideally it should be as short as possible, minding that you must guarantee the nobles' demands are met.
In light of all the above, how many days should the tourney of 1,000 princes last?
Good luck, fair sage. ;)