#April 4, 2022, C20.txt
Help20:=proc(): print(` P12(n), PR(n,R), LP(N,R) `):end:

#PILE Of pennies , so the STATE of the game is the "number of pennies"
#LEGAL MOVE: REMOVE ONE OR TWO PENNIES
#YOU LOST IF there is no pennies: 
#
#******
#***
#WE WANT A FUNCTION THAT INPUTS n and outputs 1 if is a winning position and 0 if it a losing position
#ALSO THE WINNING MOVE
#
#P12(n): 1-pile Nim with {1,2} removal
P12:=proc(n) local i,S:
option remember:



S:={}:

for i from 1 to min(2,n) do
 if P12(n-i)[1]=0 then
   S:=S union {i}:
 fi:
od:
if S={} then
 0,S:
else
 1,S:
fi:

end:

#PR(n,R): Inputs a pos. integer n and a set of POSITIVE integers R
#outputs 0,{} or 1,S where 0 means losing and S is the set of winning moves
PR:=proc(n,R) local i,S:
option remember:


S:={}:

for i in R do
 if i<=n and PR(n-i,R)[1]=0 then
   S:=S union {i}:
 fi:
od:
if S={} then
 0,S:
else
 1,S:
fi:

end:
#
#LP(N,R): Inputs a pos. integer N and a removal set R outputs the
#list of losing positions <=N
LP:=proc(N,R) local L,i:
L:=[]:
for i from 1 to N do
 if PR(i,R)[1]=0 then
  L:=[op(L),i]:
 fi:
od:
L:
end: