Ponder This

"But what if I don't have that much money?"

Just run a deficit. Does that bother you?

"But what if my neighbor is running a deficit on January 1?"

If your neighbor was running a ten dollar deficit on January 1, then on April 15 you give the government a ten dollar deficit. That's the same as if the government pays you ten dollars.

Sam: "I don't have a neighbor, since I'm at position 0."

That's okay, Sam, you alone don't have to pay taxes. Just pretend your non-existent neighbor has zero net worth.

"What if people change places, drop out, come into the system...?"

None of that is allowed. Furthermore, no money ever changes hands except for payment of taxes. Okay, no further questions.

"Wait, wait. Why are we doing a US-centric April 15 tax day, if IBM is an international company?"

I said no further questions.

Sam starts out with one dollar, and the rest of you start with varying amounts. Some of you will start with no money, in particular the four people on the far right-hand end of the line will be broke. Some of you might even start with deficits.

...

Four years pass. At the end of those four years, Sam still has his one dollar, and the only other people with nonzero net worth are at positions q, r, s and t, with q < r < s < t. Here q,r,s,t are unknown integers, not assumed to be consecutive, and the answer to the problem will involve only the values q,r,s,t.

How much did the government collect in the first year (net)? (That is, collections minus payouts.)