memo={} def naive(n,c0,c1,d0,d1,f): if n==0: return f==0 if (c0,c1,d0,d1,f) in memo: return memo[(c0,c1,d0,d1,f)] mex=set() if c0>0 and d0>0: mex.add(naive(n-1,c0-1,c1,d0-1,d1,f^1)) if c1>0 and d0>0: mex.add(naive(n-1,c0,c1-1,d0-1,d1,f^1)) if c0>0 and d1>0: mex.add(naive(n-1,c0-1,c1,d0,d1-1,f^1)) if c1>0 and d1>0: mex.add(naive(n-1,c0,c1-1,d0,d1-1,f)) g=0 while g in mex: g+=1 memo[(c0,c1,d0,d1,f)]=g return g def check(n,c0,c1,d0,d1,f): if min(c0,c1,d0,d1)==0: if c0==0: sm=d1%2 elif c1==0: sm=0 elif d0==0: sm=c1%2 else: sm=0 return (sm&1)==f if n%2==1: return True if min(c0,c1,d0,d1)==1: return True return False if 1: for n in range(1,9): for c0 in range(n+1): for d0 in range(n+1): for f in range(2): c1=n-c0 d1=n-d0 #print((n,c0,c1,d0,d1,f),naive(n,c0,c1,d0,d1,f)) # if naive(n,c0,c1,d0,d1,f)==0: # if min(c0,c1,d0,d1)>=1: # print("lose",n,(c0,c1,d0,d1,f)) # elif min(c0,c1,d0,d1)>=1: # print("win",n,(c0,c1,d0,d1,f)) ans1=(naive(n,c0,c1,d0,d1,f)>=1) ans2=check(n,c0,c1,d0,d1,f) if ans1!=ans2: print(n,(c0,c1,d0,d1,f),ans1,ans2) def binom(n,r): return int((n&r)==r) def count(n): c=0 for i in range(60): if (n>>i)&1: c+=1 return 1<