Understanding reinvestment in a miner growth scenario
I will not give you the straight out answer, but i will guide you there.
Basically the question is asking when will Eve have 51% of the network hashing power.
Please note that in this question we dont really need to deal with hashing power, this question can be answered by calculating when Eve will have more than $4M+ worth of hashing hardware.
There is an easy approximation, and then there is a more difficult, but more precise way to calculate this.
Currently everyone has $1,000,000 of hashing power. So everybody has a 1 in 5 chance of finding a block.
Lets assume that Eve will get every 5th block (as she accumulates more hashing power, she will have a higher probability of finding new blocks - incorporating this into the answer is the more difficult but precise way).
So Eve will get 1/5 of all blocks, In a day she will get 144/5 = 28.8 blocks.
28.8 blocks @12.5 BTC @ $500 = $180,000
Now we need eve to have 51% of all hashing power. Or more than the others combined, so more than $4M
So Eve needs to spend at least $3M on equipment.
$3M / $180,000 = 16.67 days
So the approximation is 16.67 days, but Eve will get there faster because as she ads more equipment, her chances of finding new blocks grow.
For the more precise calculation we need to settle on the interval of Eve getting new hardware (hourly/daily/weekly/monthly etc.)
You can map it out and eventually you will arrive at a formula.
Hope this helps.