Building the World's Tallest Structure in Record Time!

[arglebargle]

A crazed billionaire has a lifelong dream to build the world's tallest structure as a testament to his glory. Fearing competition, he decides that his tower must be far taller than any other in order to remain that way for as long as possible. The goal: build a 10 km tower, and do it in as little time as possible.

At his disposal are 1 m high x 100 m wide square blocks and special cranes/lifters. These lifters are capable of lifting as many as 5000 stacked blocks at a time and placing them on top of another stack. However, there are some limitations.

Placing a block on top of another block takes 1 week.
Similarly, placing a stack on top of another stack also takes 1 week.
However, if either of those stacks is taller than 100 m, this process takes 2 weeks.
Naturally, stacking a group of blocks or stacks precludes the possibility of doing anything else to that group during the same week (one operation per block or stack per week).

Our billionaire will spare no expense in the construction of this tower. Thus, the number of lifters and blocks is unlimited.

Using these unlimited resources, what is the smallest number of weeks in which the tower can be completed?

Spoiler

20 weeks

What method should be used to achieve this goal?

 

[davee123]

Spoiler

I get 20 weeks with a 12.288 km tower.

Week 0-1: 6144 lifters put 1 blocks onto 1 blocks, resulting in 6144 piles of 2 blocks
Week 1-2: 3072 lifters put 2 blocks onto 2 blocks, resulting in 3072 piles of 4 blocks
Week 2-3: 1536 lifters put 4 blocks onto 4 blocks, resulting in 1536 piles of 8 blocks
Week 3-4: 768 lifters put 8 blocks onto 8 blocks, resulting in 768 piles of 16 blocks
Week 4-5: 384 lifters put 16 blocks onto 32 blocks, resulting in 384 piles of 32 blocks
Week 5-6: 128 lifters put 32 blocks onto 32 blocks, resulting in 128 piles of 64 blocks and 128 piles of 32 blocks
Week 6-7: 128 lifters put 32 blocks onto 64 blocks, resulting in 128 piles of 96 blocks
Week 7-8: 64 lifters put 96 blocks onto 96 blocks, resulting in 64 piles of 192 blocks
Week 8-10: 32 lifters put 192 blocks onto 192 blocks, resulting in 32 piles of 384 blocks
Week 10-12: 16 lifters put 384 blocks onto 384 blocks, resulting in 16 piles of 768 blocks
Week 12-14: 8 lifters put 768 blocks onto 768 blocks, resulting in 8 piles of 1536 blocks
Week 14-16: 4 lifters put 1536 blocks onto 1536 blocks, resulting in 4 piles of 3072 blocks
Week 16-18: 2 lifters put 3072 blocks onto 3072 blocks, resulting in 2 piles of 6144 blocks
Week 18-20: 1 lifter puts 6144 blocks onto 6144 blocks, resulting in 1 pile of 12288 blocks

DaveE

 

[arglebargle]

Hi Dave,

Clever, but remember that the lifters are only capable of lifting up to 5000 blocks at a time. That makes your last step (last two weeks) invalid.

 

[davee123]

Clever, but remember that the lifters are only capable of lifting up to 5000 blocks at a time. That makes your last step (last two weeks) invalid.

Ooops, sorry, missed that bit... Solution is similar, though-- instead of building the 6144 blocks, they build 5000 blocks, which is easily feasible since it's smaller. Just have to do the logistics backwards to figure out the details.

DaveE