No answer so far. This is the way I think the game goes as the bag is passed from beggar to beggar.
All five beggars start with the same number of pennies, say N. Lets call the five A, B, C, D, and E, knowing that A=1, B=2 and so on, because it will get confusing using numbers for the beggars and for the pennies they put into and take out of the bag.
So A puts N pennies into the bag and passes it to C.
C takes out 2 pennies and gives to B who now has N+2 and takes out 4 pennies and gives to D who now has N+4.
Right so far. But a corollary to hint 2 is don't worry about what is in each player's pile, since they can draw from other's piles (except A's) as needed). The piles of A, B, C, and D are common property. There's no harm in tracking individual piles, and it might be interesting once you get the answer as a confirmation, but it's not necessary.
C then puts 1/2N pennies into the bag and passes it to E;
Just to clarify the notation, I think you saying that C puts (1/2)N, or N/2 pennies in the bag. That is correct.
C now has N-1/2N = 1/2N pennies and the bag has N-6 +1/2N = 3/2N -6 pennies.
Correct, and your calculations on through D getting the bag, with 27N/8-25 pennies is all correct.
Then you say:
If N =72 then at this point B will have 0 pennies and the bag will have 243. If N is less than 72 then B will have had to borrow extra pennies from C, D, or E. We also know that after D completes his turn he will pass the bag to A, and A will have only 2 pennies, which could be the end of the game. What seems confusing to me (and I may already be confused by keeping track of what is in the bag after each turn) is that when D receives the bag he takes out pennies and gives them to C and takes out 5 pennies and gives to E. So even if B had to borrow pennies from C, D, and E, C and D will still have some pennies, which contradicts the story that A leaves with the bag plus his 2 pennies and all the rest have none.
But D will have to come up with (27N/8-25)/2 pennies to put in the bag before passing it to A. Doing so will wipe out all other player's holdings, other than A's