Help := proc()

if args = NULL then
print(`Mex,NS`);
print(`NimHeap,WK,WYTQ`);
print(``);
print(`Followed Winning Ways chapter 3`);
print(`Mex = minimal excluded number`);
print(`NS = Nim Sum , NimHeaps = Nim of general number of heaps`);
print(`WK = White Knight, WYTQ = Wythoff Queen`);
print(``);
print(``);
print(``);

elif args = `Mex` then
print(`Mex(S), input the set of integer S`);
print(`return the smallest nonnegative integer that is not in S`);

elif args = `NS` then
print(`NS(a,b), input a,b the number of coin in heap1 and heap2`);
print(`return the nim-value of this game`);

elif args = `NimHeaps` then
print(`NimHeaps(S), input set of integers which are the number of bean`);
print(`in each pile`);
print(`return the nim-value of this game`);

elif args = `WK` then
print(`WK(a,b), input the position of the knight on the chess board`);
print(`The rule is you can move the knight downward or leftward or`);
print(`pick the stone of the pile (start with many stones)`);
print(`if you have to move th knight to (1,1),(1,2),(2,1) or (2,2)`);
print(`then you lose`);
print(`this function return the Grundy number of this game`);

elif args = `WTYQ` then
print(`WTYQ(a,b), input the position of the queen on the chess board`);
print(`Two players take turn moving the queen toward (0,0)`);
print(`the legal move is down or left or downleft`);
print(`player who have to move to (0,0) lost`);
print(`this function return the Grundy number of this game`);
fi:

end:


Mex := proc(S) local A,i:

A := sort([op(S)]);
A := [op({op(A)})];

if A[1] < 0 then 
   A := [op({op(A)} minus {seq(i,i = A[1]..-1)})];
fi:

for i from 0 to nops(A)-1 do
     if A[i+1] <> i then
        return(i):
     fi:
od:
return(nops(A));
end:

NS := proc(a,b)  option remember: local i,x,y,sol:
x := convert(a,base,2):
y := convert(b,base,2):
x := [op(x),0$(nops(y)-nops(x))];
y := [op(y),0$(nops(x)-nops(y))];
return(add( 2^(i-1)*(x[i]+y[i] mod 2), 
i = 1..nops(x)));
end:

NimHeaps := proc(S) option remember: local A,N,i,j:
if nops(S) = 2 then return(NS(S[1],S[2])); fi:

N:= [];
A := sort([op(S)]);
for i from 1 to nops(A) do
   for j from 1 to A[i]-1 do
       N := [op(N), NimHeap([op(1..i-1,A),j,op(i+1..nops(A),A)])];        
   od:
   N := [op(N), NimHeap([op(1..i-1,A),op(i+1..nops(A),A)])];    
od:
return(Mex(N));
end:

WK := proc(a,b) option remember: 
if a < 0 or b < 0 then return(ERROR); fi:
if a <= 2 and b <= 2 then return(0); fi:  
 
if a = 1 then return(Mex([WK(a+1,b-2)]));
elif a = 2 then
   return(Mex([WK(a+1,b-2),WK(a-1,b-2)]));
elif b = 1 then
   return(Mex([WK(a-2,b+1)]));
elif b = 2 then
   return(Mex([WK(a-2,b+1),WK(a-2,b-1)]));
else
   return(Mex([WK(a+1,b-2),WK(a-1,b-2),WK(a-2,b+1),WK(a-2,b-1)]));
fi:

end:

WYTQ := proc(a,b) option remember: local i;
if a <0  or b<0 then return(ERROR); fi:
if a = 0 and b=0 then return(0); fi:
   
return(Mex([seq(WYTQ(i,b),i=0..a-1),seq(WYTQ(a,i),i=0..b-1),
      seq(WYTQ(a-i,b-i),i=1..min(a,b))]));
end: