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?

We can see that the number of blocks produced is the same as the number of crews employed and also the number of slave deaths.  The maximum number of blocks that could be produced before work has to cease as a result of insufficient slave labor is determined as:

Inital blocks (35,000/17) = 2,058/(1 - (16/17)) = 35,000

The number of blocks in a square pyramid of N layers is detemined as (N)(N + 1)(2N + 1)/6 <=35,000.  Since 35,000 does not correspond to an integer number of layers, we must select the closest, lower number.  With help from Excel, this turns out to be 46 layers with a total of 33,511 blocks.

Therefore, the height is 46 x 3 = 138 meters.  The number of blocks required is 33,511.  And the remaining slaves are (35,000 - 33,511) = 1,489.