Rameses wishes to build a great pyramid for his interment.

The structure will have a square base and be solidly composed of cubical stone blocks. Each level of the pyramid contains one block fewer per side as the pyramid rises.

Rameses has available an initial work force of 35,000 slaves. Each morning the available labor pool is divided into work crews of 17 slaves each. Any remainder that cannot form a full crew gets the day off but are available the following day. Each crew can lay one block of the pyramid each day.

Unfortunately, the heat of the desert sun causes the death of one member of each crew each day. Work ceases on the project when it can be determined that there will be insufficient slaves available to raise the pyramid one more level. Each stone block measures 3 meters per side.

How many days will it take to construct Rameses' pyramid? How tall will it be? How many of the original slaves survive the construction?

Well, I cheated. Instead of finding an equation for the number of surviving slaves at the beginning of the nth day, or the equation for the number of blocks placed on (or up to) the nth day, I just made an excel table. So here was my basic approach.

I determined the number of slaves at the beginning of the day, the number of actual crews (which equals the number of blocks placed, and also equals the number of slaves that died), and the number of slaves at the end of the day.

From this I found that on the 136th day there would be 17 slaves, at the end of the day one would die, and then there wouldn’t enough slaves to make a crew. So I knew this was a limit point… the slaves will probably have to stop sometime before that, but at the very least I know they will stop after the 136th day at the latest.

Ok, if they worked all they way through to the end of the 136th day, they would have placed 34,984 blocks. Well, let’s find the greatest n such that "the sum of k^2 from k=1 to k=n" is less than 34,984. At n = 46 we get 33,511 and at n = 47 we get 35,720. Ok, so if they worked until the 136th day, they would have placed 1473 extra blocks for no reason. So let’s back it up, and find the last day that the total number of blocks is still less than 33,511.

That is day 52. At the end of day 52 they have placed 33,496 blocks, and 1504 slaves left. They still need 15 more blocks, though to finish the pyramid.

So if I am assuming correctly, on day 53 they would only form 15 crews. Even though they have the capacity to form 88 crews, there is no point as they only need 15. So at the end of day 53 only 15 slaves die.