# A Great Pyramid

THUDandBLUNDER
Posers and Puzzles 06 Jun '05 16:40
1. 06 Jun '05 16:40
Tutankhamun wants to build a Great Pyramid for his after-life. It must be composed of cubical stone blocks and its base must be square. Each level of the Pyramid will be 1 block shorter per side than the level below it. Tutankhamun has available a workforce of 35,000 slaves. Each morning the available labor pool is divided into workgroups of 17 slaves each. Any remaining slaves that cannot form a full workgroup will get the day off, but are available for work the following day. Each workgroup can lay 1 block of the Pyramid each day. However, the pitiless desert heat causes the death of 1 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 the construction?

3) How many days will it take to construct the Pyramid?
2. PBE6
Bananarama
06 Jun '05 18:06
Originally posted by THUDandBLUNDER
Tutankhamun wants to build a Great Pyramid for his after-life. It must be composed of cubical stone blocks and its base must be square. Each level of the Pyramid will be 1 block shorter per side than the level below it. Tutankhamun has available a workforce of 35,000 slaves. Each morning the available labor pool is divided into workgroups of 17 slaves ...[text shortened]... aves will survive the construction?

3) How many days will it take to construct the Pyramid?
The pyramid can't contain more than 35,000 blocks because slaves are dying everyday and you need at least 17 slaves to put one up. Using Excel, my favorite slave (although not the prettiest), I found that the largest sum of squares less than 35,000 is 33,511, which is the sum of 1^2 through 46^2. This is the maximum number of blocks which can be put down according to the rules, so the height of the pyramid will be no more than 46 levels.

I found that if you maximize the number of work groups everyday (except for the last day, where less than the full number of blocks needs to be placed), the 46-level pyramid is completed in 53 days with 1489 slaves surviving. Funnily enough, this is the same number of survivng slaves I got if I let 1 work group have the day off, 2 work groups have the day off, half the work groups have the day off, etc... The only thing that changes is the construction time gets longer. So I believe the final answers are:

1. 46 levels high.
2. 1489 slaves will survive.
3. The minimum time of construction is 53 days.

Upon further reflection, 35000 - 33511 = 1489, so I guess it makes sense that if one slave dies in every work group every day, then no matter how long it takes to construct the pyramid one slave will die for every block placed.

Interesting puzzle!
3. 06 Jun '05 18:471 edit
Originally posted by PBE6
The pyramid can't contain more than 35,000 blocks because slaves are dying everyday and you need at least 17 slaves to put one up. Using Excel, my favorite slave (although not the prettiest), I found that the largest sum of squares less than ...[text shortened]... one slave will die for every block placed.

Interesting puzzle!
my number of survivors is slightly different:

46 levels high
1416 survivors
53 days elapse time

Very hard to avoid calculation errors (should program it)

edit. my miscalc. It is 1489 survivors indeed.
4. The Plumber
Leak-Proof
06 Jun '05 19:15
Originally posted by THUDandBLUNDER
Tutankhamun wants to build a Great Pyramid for his after-life. It must be composed of cubical stone blocks and its base must be square. Each level of the Pyramid will be 1 block shorter per side than the level below it. Tutankhamun has available a workforce of 35,000 slaves. Each morning the available labor pool is divided into workgroups of 17 slaves ...[text shortened]... aves will survive the construction?

3) How many days will it take to construct the Pyramid?
An interesting problem. Let me ask a question.... Since it takes a 17-slave workgroup a day to put a stone in place, can we assume that only one level can be completed in a day? If that's the case, I think PBE6 and Mephisto need to rethink their solutions....
5. PBE6
Bananarama
06 Jun '05 20:21
Originally posted by The Plumber
An interesting problem. Let me ask a question.... Since it takes a 17-slave workgroup a day to put a stone in place, can we assume that only one level can be completed in a day? If that's the case, I think PBE6 and Mephisto need to rethink their solutions....
I think you're referring to this scentence from the original question:

"Work stops on the Pyramid when it can be shown that there will not be sufficient slaves available to raise the Pyramid another level."

I took this to mean that if it is clear that no more than 16 slaves will be left before the next level is complete, then work stops (which is an odd thing to say anyway, because the base would be built first and would define the total height of the pyramid). But if we assume work stops after each level is complete, then the construction time would be longer. Let's see...<crunch> <crunch> <crunch>...<crunch> <crunch> <crunch>...

Hmm...this is going to take a little longer than I thought. I'll get back to y'all.

6. The Plumber
Leak-Proof
06 Jun '05 20:27
Not exactly - I was reading "common sense" into the problem (put in quotes because common sense is anything but common, and often nonsensical). I was just reading that if it takes a day for the slaves to put a stone in place, then the guys who are putting the stone that goes on top of stone A can't start today, because stone A isn't in place yet. Your previously stated solution actually has them putting more stones in place on day one than can fit in level one.

I may be trying to be too literal here, but it does make the problem more interesting.... (are you done crunching yet) ðŸ˜€
7. PBE6
Bananarama
06 Jun '05 20:36
Originally posted by The Plumber
Not exactly - I was reading "common sense" into the problem (put in quotes because common sense is anything but common, and often nonsensical). I was just reading that if it takes a day for the slaves to put a stone in place, then the guys who are putting the stone that goes on top of stone A can't start today, because stone A isn't in place yet. Y ...[text shortened]... literal here, but it does make the problem more interesting.... (are you done crunching yet) ðŸ˜€
I can see that. However, I could also see exhaustion being the limiting factor on how many stones per day a group can place. In that case, group A might finish before "lunch" (which of course would consist of whippings and gruel), and group B could place the stone after "lunch".

And speaking of lunch, I'm done with the crunch. Here's what I got:

It would now take 82 days to complete the 46-level pyramid, with 1489 slaves surviving the ordeal. This is about 55% longer than before.
8. 06 Jun '05 21:02
Originally posted by PBE6
I think you're referring to this scentence from the original question:

[b]"Work stops on the Pyramid when it can be shown that there will not be sufficient slaves available to raise the Pyramid another level."

I took this to mean that if it is clear that no more than 16 slaves will be left before the next level is complete, then work stops (which i ...[text shortened]...
Hmm...this is going to take a little longer than I thought. I'll get back to y'all.

[/b]
"which is an odd thing to say anyway, because the base would be built first and would define the total height of the pyramid"

Actually, I think the pyramid can be constructed differently as well: first a square 2x2 and a top above it. That are two 'complete' layers. Then 12 blocks around the 4 and 8 around the 1 and then reconstruct the top 2 layers above it. Etc... Each time we reconstruct the top , we have a full pyramid.
9. PBE6
Bananarama
06 Jun '05 21:09
Originally posted by Mephisto2
"which is an odd thing to say anyway, because the base would be built first and would define the total height of the pyramid"

Actually, I think the pyramid can be constructed differently as well: first a square 2x2 and a top above it. That are two 'complete' layers. Then 12 blocks around the 4 and 8 around the 1 and then reconstruct the top 2 layers above it. Etc... Each time we reconstruct the top , we have a full pyramid.
Hmm, that's true. It could be constructed that way. I'm not going to try calculating how long it would take for them to stop after each layer, or after each "composite" layer is complete, though. My abacus would catch fire!!!
10. 06 Jun '05 22:13
how do only 1489 slaves survive after 53 days if the desert only kills 1 every day?
11. 06 Jun '05 23:20
Originally posted by fredthebug
how do only 1489 slaves survive after 53 days if the desert only kills 1 every day?
One per workgroup per day.
12. 07 Jun '05 00:05
oh... i get it. that explains why i couldnt get the puzzle
13. 08 Jun '05 08:33
It would now take 82 days to complete the 46-level pyramid, with 1489 slaves surviving the ordeal. This is about 55% longer than before.
I don't understand your logic - how can it take more days without killing more slaves?
14. 08 Jun '05 09:00
Originally posted by thundyr
I don't understand your logic - how can it take more days without killing more slaves?
the number of slaves killed only depends on the number of cubes (17 people needed to put one in place, one of them dies), not on the organization and elapse time of the pyramid.