回溯——3、5升杯倒4升水

接前面

递归的改造——间隔挑硬币打印所挑选的硬币需要用到回溯。但书上的回溯没说一定要用递归调用。

书上说

回溯的本质是深度优先，还需要建立栈。

数学浅谈

一般说用扩展欧几里得除法的方法。

最大公约数

小学数学也要学复习。这里理解就是3升水和5升水都可以有1升水组成。4升和6升最大有两升组成。

$gcd(a,b)=x\ast a+y\ast b$

拙见最大公约数1升的，可以倒出所有1的倍数的水，最大公约数2升的，可以倒出所有2的倍数的水。
[基础定理]任意给定二个整数a,b, 总是存在二个自然数x, y, 使得$gcd(a,b)=x\ast a+y\ast b$.这里要给数学家来点难度，只给一杯8升水怎么倒？

$P_3^2=6$

三杯里选两杯有6种情况。C是组合，这里是排列，搞错了。排列的英文是 Permutation 或者 Arrangement，因此在数学符号中，用 P 或者 A 表示都可以，二者意思完全一样。

只要一杯8升水

有6种情形可以倒，但要i杯有水且j杯有空间，才能i倒j，可以i倒满j，或者i倒空j未满出，不能洒；但这个倒法不能重复之前的状态,否则无限循环；直到最后有一杯出现了4升水。

代码

def next_step(num):
    for i in range(num):
        for j in range(num):
            if i != j:
                yield i, j
def isin(a, b):
    for i in b:
        if a == i:return True
    return False   
                 
def f(s, trace, X=[], d=4):
    #准备初值：s原始杯子状态二维列表,trace倒的步骤三维数组第一步默认状态[0,5], [8,8], [0,3],X解正确的步骤,d目标水升数
    global traceback
    traceback += 1
    for k in s:
        if d == k[0]:#最后的成功,倒出d了。
            X.append(copy.deepcopy(trace))
            return True
    #准备初值：P(3,2)三杯里选两杯有6种情况    
    a = next_step(len(s))
    for i, j in a:#(范围未超界并且工作未完成)
        #分析条件：i杯有水且j杯有空间，才能i倒j
        if 0 != s[i][0] and 0 != s[j][1] - s[j][0]:#保证满足条件才往下走
            a = s[i][0]
            b = s[j][0]
            if s[i][0] >= s[j][1] - s[j][0]:#分析条件：i倒满j
                s[i][0] = s[i][0] + s[j][0] - s[j][1]
                s[j][0] = s[j][1]
            else:#分析条件：i倒空j未满
                s[j][0] += s[i][0]
                s[i][0] = 0
            if isin(s, trace):#本层更换选择;横向走
                #不能有重复状态,否则无限循环,i,j要恢复分析条件前的状态。
                s[i][0] = a
                s[j][0] = b
                continue 
            else:#(成功)局部成功，可以下一步。 
                trace.append(copy.deepcopy(s))#路径进栈;由第一选择开始进入下一层;纵向走
            f(s, trace, X, d)#纵向走 下一层
            #(工作未完成)弹出堆栈；原来的上一层更换为下一选择
            trace.pop()#回溯到上一层横向走。前面的for那里。
            #这里i,j也要恢复分析条件前的状态。
            s[i][0] = a
            s[j][0] = b
    #(全部工作未完成)因为有时六种倒法都不对。
    return X #输出结果

一般回溯方法的程序结构

把书上的每一步都写到代码里，没有什么能比这个说得再清除了。

打印

总共有13种倒法。总共回溯了212次。

>>> b=f([[0,5], [8,8], [0,3]], [[[0,5], [8,8], [0,3]]], [], 4)
>>> for x in b:print(b)

[[[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]]]
[[[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]]]
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[[[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[5, 5], [0, 8], [3, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[0, 5], [7, 8], [1, 3]], [[1, 5], [7, 8], [0, 3]], [[1, 5], [4, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [0, 8], [3, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]], [[[0, 5], [8, 8], [0, 3]], [[0, 5], [5, 8], [3, 3]], [[3, 5], [5, 8], [0, 3]], [[3, 5], [2, 8], [3, 3]], [[5, 5], [2, 8], [1, 3]], [[5, 5], [3, 8], [0, 3]], [[2, 5], [3, 8], [3, 3]], [[2, 5], [6, 8], [0, 3]], [[0, 5], [6, 8], [2, 3]], [[5, 5], [1, 8], [2, 3]], [[4, 5], [1, 8], [3, 3]]]]
>>> print(traceback)
212
>>> print(len(b))
13
>>>