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Title: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 6th, 2003, 5:18am Tutankhamun wishes to build a Great Pyramid for his after-life. It must be solidly composed of cubical stone blocks and its base must be square. Each level of the Pyramid will be one block shorter per side than the level below it. Tutankhamun has available an initial workforce of 35,001 slaves. Each morning the available labour pool is divided into workgroups of 17 slaves each. Any remaining slaves that cannot form a full workgroup get the day off, but are available for work the following day. Each workgroup can lay one block of the Pyramid each day. However, the merciless desert heat causes the death of one member of each workgroup per day. Work stops on the Pyramid when it can be shown that there will not be sufficient slaves available to raise the Pyramid another level. 1) How many levels will it have? 2) How many of the original slaves will survive its construction? 3) How many days will it take to construct the Great Pyramid? |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 6th, 2003, 5:44am This is really one of those problems where you shouldn't make it more complicated for yourself than you have to ::) Now if only I had realized that a few minutes sooner.. nice puzzle.. btw, can we assume the slaves that aren't needed on the last day are also off work, and thus don't die doing nothing in a workgroup |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 6th, 2003, 6:15am Quote:
Yes, although this is not really relevant to the solution. |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 6th, 2003, 7:04am Well, it's relevant to how many survive, because if they were to form workgroups 1 in 17 of them would die. |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 6th, 2003, 8:09am The following two quotes seem to be incompatible: Quote:
Quote:
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 6th, 2003, 8:52am It may seem so, but it isn't.. It is slightly less complicated to just have all people on the last day in workgroups and have 1/17th of them die, but it makes less sense.. And more importantly both ways give different answers, independant of how complicated you make the solution process.. But if you want it as least complicated as possible I'll just have all the slaves killed on the final day: 0 survivors, easy :P |
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Title: Re: Tutankhamun's Pyramid Post by aero_guy on Nov 6th, 2003, 9:18am Is there any easier way to solve part one than[hide] summing squares until you exceed total slaves[/hide]? And is there a quicker way to solve the last part than[hide] iteratively dividing and subtracting slaves and bloakcs[/hide]? |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 6th, 2003, 9:40am on 11/06/03 at 07:04:34, towr wrote:
Not necessarily. Because of the constraints of the problem, the number of survivors determines how many were redundant at the end, not vice versa. I am not even sure if the latter number can be calculated from the given info. Quote:
Yes. Quote:
No, not unless towr's ruminations on redundant survivors can simplify matters. |
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Title: Re: Tutankhamun's Pyramid Post by Icarus on Nov 6th, 2003, 4:41pm Having pulled up Excel and put it in, I too quickly discovered - but too late, towr's shortcut. An equivalence that is stated in the problem, but unseen until I saw the maximum # of blocks the workers could place without regard to layers. Quote:
??? I come up with [hide]924[/hide] redundant workers on the last day. Its not hard to figure out. And the way I approached the problem (probably much as towr did), you find it, and the number of survivors among the unlucky last-day workers to obtain the total survivors of the process. I'm sure the real pyramid buiders (Tutankaton was not a part of the pyramid building period) would have loved such productive slaves! The real pyramids took years to build, not months! |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 7th, 2003, 3:17am The three best known pyramids were a bit larger I think.. And they weren't created by slaves, but a normal workforce that went on strike when there wasn't enough make-up :P (really, records of this have been found) But Tutankhamun didn't have a pyramid build anyway afaik.. on 11/06/03 at 16:41:50, Icarus wrote:
Let's check our numbers ::[hide]One worker dies for every block, so we're looking for the maximum number of levels n for which n(n + 1)(2n + 1)/6 < 35001-16 This gives n=46. The amount of days it takes to build the pyramid is 53. At the start of the last (53th) day I have 1504 slaves left, who need to lay 14 blocks. So if the 1266 redundant slaves get off, just 14 workers die, and there are 1490 survivors in total. [/hide]:: |
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Title: Re: Tutankhamun's Pyramid Post by Icarus on Nov 7th, 2003, 3:57pm I interpreted the phrase "Each level of the Pyramid will be one block shorter per side than the level below it" differently than you did. I took it to mean that all the way around the pyramid each level was one block further in than the level below. So the number of blocks per side would be 2 less than the level below. Consequently, I was summing the squares of the odd integers only, not all of them. I threw my numbers away, so I don't have them, and don't have time to recreate them right now. But I may put them up later. Tutankhamun, who started his reign as Tutankhaton, only lived to be 18. Further, pyramid building only took place a period in the Old Kingdom. Tutankhamun reigned during the Middle Kingdom, I believe. So he had a tomb. not a pyramid. Further, Tutankhamun got caught in a religious contraversy. His uncle and predescessor, Ahkenaton (almost certainly mispelled - I'm rushing) broke away from the traditional Egyptian belief system, demanding that there was only one god: the sun god Aton. He set up temples to Aton, and shut down those of the other gods. Apparently he had a great influence on his nephew, who became Pharoah on his death. Tutankhaton started his reign in favor of Aton worship, but was too young to effectively hold power like his uncle had. Eventually he caved to the traditionalists who were trying to put down the Aton heresy and return to the traditional religion. He even changed his name to Tutankhamun, in honor of Amon Ra. With his power mostly stripped away, he ended up dying as a teenager, and another uncle - a staunch traditionalist - succeeded him. This uncle immediately set about destroying every trace of Aton worship, trying to remove all evidence that it ever existed. Part of this was the obliteration of all external sign of Tutankhamun's small tomb, and any mention of his or of Akhenaton's reign. The irony is of course that by doing so, he set things up for Tutankhamun to become the most famous of Pharoahs in modern times. Since Tutanhamun's tomb was hidden, and his reign unknown by ancient tomb robbers, his inner tomb was never looted until Howard Carter got a crack at in the 1920s. His is the only unlooted tomb of a Pharoah ever found. |
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Title: Re: Tutankhamun's Pyramid Post by Icarus on Nov 7th, 2003, 8:53pm Found a quick reference for Tutankhamun's Life (http://homepage.powerup.com.au/~ancient/tut2.htm). I was wrong - he was part of the New Kingdom. Still nowhere near the Pyramid builders. He is now thought to have been the son of Akhenaten with a lesser wife, rather than a nephew. And his immediate successor was a counselor (and possible step-grandfather) who only reigned 4 years before power was seized by Horemheb, who was the commander of the armies. It was Horemheb to attempted to obliterate all traces of his three predecessors, and thus inadvertently made Tutankhamun into the most famous of all Pharoahs. |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 10th, 2003, 4:31am on 11/07/03 at 03:17:34, towr wrote:
How did you get these numbers? |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 10th, 2003, 5:04am I used a single line MOO-program. (And as such I don't have it anymore, since I only use single line programs on the commandline. I can however create a similar program easily enough.) ;if(1) a=35001; c=0; d=0; while (c < 33511) b = min(a/17, 33511-c); c = c+ b; d =d+1; player:tell(d, " ", a, " ", b, " ",c); a=a-b; endwhile endif summary of the result: ::[hide]column 1) day column 2) slaves at the start of the day column 3) blocks laid = slaves died at end of the day column 4) total block in pyramid at the end of the day 1 35001 2058 2058 2 32943 1937 3995 3 31006 1823 5818 4 29183 1716 7534 5 27467 1615 9149 ..... 50 1802 106 33305 51 1696 99 33404 52 1597 93 33497 53 1504 14 33511 [/hide]:: |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 19th, 2003, 4:24am Quote:
A Pavlovian grab for the keyboard is not required in order to answer 3). :P Let the number of days required = n One may obtain a lower bound for n by noting that we need (16/17)n >= 1490/35001 This gives n >= 52.068, and indeed 53 days are enough to lay 33511 blocks. |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 19th, 2003, 4:43am So how do you proof 53 days is enough? Anything higher than the lower bound might have been the right answer.. Without an upperbound you still have to go through all the days to say where you end up, don't you? |
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Title: Re: Tutankhamun's Pyramid Post by THUDandBLUNDER on Nov 19th, 2003, 10:17pm Quote:
The worst-case scenario is when 16 slaves are idle every day of the construction. In that case the number of blocks laid in n days will be 35001*[1 - (16/17)n] - 16n/17 Substituting n = 53 gives number of complete blocks laid = 33542 > 33511 |
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Title: Re: Tutankhamun's Pyramid Post by towr on Nov 20th, 2003, 12:35am nice.. I'll try to remember this next time I face a similar problem.. |
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