A classic:
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?
Personally I feel that a group could not work on the next layer up
until the previous layer was finished. On this premise build one
complete layer per day. On the first day there are a possible
2058 crews. The square of that number is 45 and some
change. Therefor at day one 2025 crews would be working on the
pyramid. 33 crews would go home along with the slaves that did
not form a crew. Using this idea and a base of 45 I ran out of
slaves at day 3 (not enough crews to build 1 complete level)
Changing the base width to 44 stones I ran out at day 8. A
complete pyramid could be build if the base was 43. Thus the
height would be 129 meters and 7566 slaves would remain.
I am sure that a maximum pyramid could be built assuming that the part
of the pyramid was not being built over a base that was still being
completed; or percentage of crews working on next level is equal to
percentage of base completed the previous day. Will have to keep
fiddling with that idea.