## My Funeral Planning.

When I die, I wish to be buried in a pyramid. A little egotiscal I know, but what the hey! The structure will have a square base and be solidly composed of cubical stone blocks. Each level of the pyramid contains one less block per side as the pyramid rises. I have gathered together an initial work force of 35,000 slaves. No mean feat in this day and age of employment law, I can tell you!

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 (my kind and caring nature got the better of me there) 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 my pyramid?

How tall will it be?

How many of the original slaves survive the construction?

June 3rd, 2014 at 8:03 am

Hi Karl. I wrote a program and found that we’re in the neighbourhood of 137 days of building. There’s no way I’m going to do a day by day account of the build. I haven’t yet found a slick trick yet. I fancy that noting that e.g. 35000 = 7*17^3 + 2*17^2 + 1*17 + 14, and am tempted to think that that sort of representation of numbers may lead to a short cut.

June 3rd, 2014 at 11:27 am

I haven’t thoroughly checked my computations. But in the interests of posting something…

So far I’ve only done this using Visual Basic (under Excel). The slaves are capable of laying 34864 blocks and that would take 137 days. However, the biggest pyramid that can be built using up to that number of blocks has a base of side 46 (that’s also the height, in blocks) and only contains 33511 blocks. That would require a mere 53 days to build. The number of remaining slaves is 35000-33511 = 1489. The height is 3*46 = 138 metres.

But as for a mathematical approach, I’m still stumped for a nice technique.

FWIW the number of blocks in a pyramid of height h (blocks) is h(h+1)(2h+1)/6 and is the sum of the first h natural squares.

June 4th, 2014 at 2:29 am

Well done Chris – Your answers are correct – here is my maths – no nice technique – just brute force and a big blackboard…

Each block added results in the death of exactly one slave, and when there are less than 17 slaves no more blocks may be added, so the pyramid consists of at most 35000-16 = 34984 blocks. The top level needs 1 block, the next 4, and in general the xth level requires x^2 blocks, and so a pyramid n levels tall requires 1 + 2^2 + 3^2 + … + n^2 = n(n+1)(2n+1)/6 blocks. This is between n^3/3 and (n+1)^3/3, and solving n^3/3 = 34984 we get that the maximum number of levels is either 46 or 47, with a quick check showing that 46 is correct.

It follows that the pyramid is 46×3 = 138 meters tall, consists of 33511 blocks, and the number of slaves surviving the construction is 35000-33511=1489.

To compute the number of days required for construction,

note that if there are m slaves on a given day the number of blocks added is between m/17 and m/17 – 16/17, and given k days it can be easily shown that the number of blocks added is between m/17 + (16/17)m/17 + … + (16/17)^{k-1} m/17 and m/17 + (16/17)m/17 + … + (16/17)^{k-1} m/17 – k*16/17; summing the geometric series we get m*(1-(16/17)^k), and so to find the number of days required we solve 35000*(1-(16/17)^k) = 33511 and get that (16/17)^k = .04254 or k=52.08. Since this came from an upper bound on the number of blocks, it’s a lower bound on the number of days; thus, it’ll take at least 53 days to build the pyramid. Checking the lower bound, we see that the number of blocks that can be added in 53 days is greater than 33542, and so the pyramid can indeed be built in 53 days.

June 4th, 2014 at 4:58 am

Aaargh. Nicely done Karl. Your answer is much better than mine.