Continuing the discussion from Hoofball Bronze February 2018:
I will come back to the linked thread about my approach later on. However, I want to understand why the official analysis is correct. It says that the only cases FJ has to distribute a ball is:
- A cow is a “source” (passes the ball away but never gets a ball back);
- A pair of adjacent cows who pass to each other and don’t receive from anyone else.
I don’t follow why these are the only 2 cases. If these conditions are met, why must it be possible for all cows to touch the ball?
What happens if a cow receives a pass from 2 cows (both neighbors) and passes back to one of them?
What about a pair that passes to each other but does not receive passes solely from its other partner?
Intuitively, why will every cow who receives a pass from A and passes to B \ne A always touch the ball if the “source” and “isolated pairs” are given initial balls?
Why is a pair of adjacent cows who receive & pass solely b/w each other the only such “isolated component”?
Why are cows with indegree of 0 and pairs with isolated components the only cows we have to consider?
I can’t wrap my head, intuitively, around why those categories are the only initial players who need balls. Thank you, in advance.