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?
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Submitted by SilverKnight
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Rating: 3.0000 (1 votes)
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Solution:
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Each block added results in the death of exactly one slave, and when there are fewer than 17 slaves no more blocks may be added, so the pyramid consists of at most 35000-16 = 34984 blocks. The first level needs 1 block, the next 4, and in general the ith level requires i^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. |
