==> competition/games/nim.s <==
For the particular game described we only need to consider positions for
which the following condition holds for each pile:

	(number of bills in pile k) + k >= (number of piles) + 1

A GOOD position is defined as one in which this condition holds,
with equality applying only to one pile P, and all piles following P
having the same number of bills as P.
( So the initial position is GOOD, the special pile being the first. )
I now claim that if I leave you a GOOD position, and you make any move,
I can move back to a GOOD position.

Suppose there are n piles and the special pile is numbered (n-p+1)
(so that the last p piles each contain p bills).
(1) You take p bills from p or more piles; 
  (a) If p = n, you have just taken the last bill and lost.
  (b) Otherwise I reduce pile (n-p) (which is now the last) to 1 bill.
(2) You take p bills from r(<p) piles;
    I take r bills from (p-r) piles.
(3) You take q(<p) bills from p or more piles;
    I take (p-q) bills from q piles.
(4) You take q(<p) bills from r(<p) piles;
  (a) q+r>p; I take (p-q) bills from (q+r-p) piles
  (b) q+r<=p; I take (p-q) bills from (q+r) piles

Verifying that each of the resulting positions is GOOD is tedious
but straightforward.  It is left as an exercise for the reader.

    -- RobH