Even if PG gave you such a task, how would that demonstrate anything relevant? You getting a task and a deadline is the opposite of being entrepreneurial; that's called having a job and a boss and showing that you are an adequate or better employee.
Being entrepreneurial involves you
1) generating ideas,
2) using your intuition to pick a good one,
3) testing it out to see if your intuition is correct (iterate if not), and
4) then execute that idea (and again, iterate as needed)
Or, in short: discovering and exploiting an opportunity
Oh, I don't know. Having a team that can solve any arbitrary difficult problem in three days could be quite useful in a startup. The (technical) execution of a startup idea can be cast as a series of arbitrarily difficult problems, after all.
My two long term pet peeves for which I would be most happy to help lobby to get them a hearing:
1) Faster than light travel
2) Create a robot that carries it's own power supply, communication device, and webcams that would allow me to traipse up and down any path in Rocky Mountain National Park in real time so that I can take a hike for 20 minutes when I need a break without the requisite plane travel and hotel bill
I suspect pg would be willing to pay for at least number 1 :-)
Well, I'd settle for a teleportation device or ability that could take me anywhere in the universe instantaneously, and far, far greater physical durability (maybe in a Culture-level gel suit).
They better get cranking, because 72 hours is just not a lot of time to develop either of these things.
Could be useful IN a startup, but not AS a startup.
AFAIK YCombinator is for creating startups, not for letting outside people be a part of one.
You want to prove your will? Ask your neighbours, parents, friends, people you find in the street, ask them for what troubles them, what little things they have to do what would rather not. Find a way to solve it, charging money, low enough that they would rather pay, high enough that you cover costs.
Then you either have a nice startup of your own or at least you have some experience that you can share here. Repeat until success or bust.
PG has argued repeatedly that the idea a startup comes up with is the least reliable and arguably least important factor in whether that startup will be successful.
If the idea doesn't matter, the method of getting the idea should matter even less. What's important is whether the team can a) respond to feedback on their idea and b) code quickly and effectively to put that response into action.
The idea matters a bit, even if it's not the top priority. If the idea doesn't matter at all, why not just take your team of gifted programmers and start a consulting company hacking and finishing all client requests in 72 hours or less? If such a company existed, they can probably charge six figures for every project they take on.
Being entrepreneurial involves you
1) generating ideas,
2) using your intuition to pick a good one,
3) testing it out to see if your intuition is correct (iterate if not), and
4) then execute that idea (and again, iterate as needed)
Or, in short: discovering and exploiting an opportunity
Edit: minor.