思路：一共4个数字，共需要3个运算符，可以构造一个二叉树，没有子节点的节点的为值，有叶子节点的为运算符
例如数字{1, 2, 3, 4}，其中一种解的二叉树形式如下所示：
因此可以遍历所有二叉树可能的形式，4个数的全排列，从4种运算符中挑选3种运算符（运算符可以重复）
核心步骤1：需要遍历所有二叉树的可能，参考Eric Lippert的方法
class Node { public Node Left { get; private set; } public Node Right { get; private set; } public Node(Node left, Node right) { this.Left = left; this.Right = right; } } static IEnumerable<Node> AllBinaryTrees(int size) { if (size == 0) return new Node[] { null }; return from i in Enumerable.Range(0, size) from left in AllBinaryTrees(i) from right in AllBinaryTrees(size - 1 - i) select new Node(left, right); }
核心步骤2：对于任意一个二叉树，构造表达式树
static Expression Build(Node node, List<double> numbers, List<Func<Expression, Expression, BinaryExpression>> operators) { var iNum = 0; var iOprt = 0; Func<Node, Expression> f = null; f = n => { Expression exp; if (n == null) exp = Expression.Constant(numbers[iNum++]); else { var left = f(n.Left); var right = f(n.Right); exp = operators[iOprt++](left, right); } return exp; }; return f(node); }
核心步骤3：遍历4个数字的全排列，全排列参考这里
static IEnumerable<List<T>> FullPermute<T>(List<T> elements) { if (elements.Count == 1) return EnumerableOfOneElement(elements); IEnumerable<List<T>> result = null; foreach (T first in elements) { List<T> remaining = elements.ToArray().ToList(); remaining.Remove(first); IEnumerable<List<T>> fullPermuteOfRemaining = FullPermute(remaining); foreach (List<T> permute in fullPermuteOfRemaining) { var arr = new List<T> { first }; arr.AddRange(permute); var seq = EnumerableOfOneElement(arr); if (result == null) result = seq; else result = result.Union(seq); } } return result; } static IEnumerable<T> EnumerableOfOneElement<T>(T element) { yield return element; }
例如有四个数字{1, 2, 3, 4}，它的全排列如下：
1, 2, 3, 4 1, 2, 4, 3 1, 3, 2, 4 1, 3, 4, 2 1, 4, 2, 3 1, 4, 3, 2 2, 1, 3, 4 2, 1, 4, 3 2, 3, 1, 4 2, 3, 4, 1 2, 4, 1, 3 2, 4, 3, 1 3, 1, 2, 4 3, 1, 4, 2 3, 2, 1, 4 3, 2, 4, 1 3, 4, 1, 2 3, 4, 2, 1 4, 1, 2, 3 4, 1, 3, 2 4, 2, 1, 3 4, 2, 3, 1 4, 3, 1, 2 4, 3, 2, 1
核心步骤4：从4种运算符中挑选3个运算符
static IEnumerable<IEnumerable<Func<Expression, Expression, BinaryExpression>>> OperatorPermute(List<Func<Expression, Expression, BinaryExpression>> operators) { return from operator1 in operators from operator2 in operators from operator3 in operators select new[] { operator1, operator2, operator3 }; }
最后是Main函数：
static void Main(string[] args) { List<double> numbers = new List<double> { 1, 2, 3, 4 }; var operators = new List<Func<Expression, Expression, BinaryExpression>> { Expression.Add,Expression.Subtract,Expression.Multiply,Expression.Divide }; foreach (var operatorCombination in OperatorPermute(operators)) { foreach (Node node in AllBinaryTrees(3)) { foreach (List<double> permuteOfNumbers in FullPermute(numbers)) { Expression expression = Build(node, permuteOfNumbers, operatorCombination.ToList()); Func<double> compiled = Expression.Lambda<Func<double>>(expression).Compile(); try { var value = compiled(); if (Math.Abs(value - 24) < 0.01) Console.WriteLine("{0} = {1}", expression, value); } catch (DivideByZeroException) { } } } } Console.Read(); }
计算结果：
(4 * (1 + (2 + 3))) = 24 (4 * (1 + (3 + 2))) = 24 (4 * (2 + (1 + 3))) = 24 (4 * (2 + (3 + 1))) = 24 (4 * (3 + (1 + 2))) = 24 (4 * (3 + (2 + 1))) = 24 (4 * ((1 + 2) + 3)) = 24 (4 * ((1 + 3) + 2)) = 24 (4 * ((2 + 1) + 3)) = 24 (4 * ((2 + 3) + 1)) = 24 (4 * ((3 + 1) + 2)) = 24 (4 * ((3 + 2) + 1)) = 24 ((1 + 3) * (2 + 4)) = 24 ((1 + 3) * (4 + 2)) = 24 ((2 + 4) * (1 + 3)) = 24 ((2 + 4) * (3 + 1)) = 24 ((3 + 1) * (2 + 4)) = 24 ((3 + 1) * (4 + 2)) = 24 ((4 + 2) * (1 + 3)) = 24 ((4 + 2) * (3 + 1)) = 24 ((1 + (2 + 3)) * 4) = 24 ((1 + (3 + 2)) * 4) = 24 ((2 + (1 + 3)) * 4) = 24 ((2 + (3 + 1)) * 4) = 24 ((3 + (1 + 2)) * 4) = 24 ((3 + (2 + 1)) * 4) = 24 (((1 + 2) + 3) * 4) = 24 (((1 + 3) + 2) * 4) = 24 (((2 + 1) + 3) * 4) = 24 (((2 + 3) + 1) * 4) = 24 (((3 + 1) + 2) * 4) = 24 (((3 + 2) + 1) * 4) = 24 (1 * (2 * (3 * 4))) = 24 (1 * (2 * (4 * 3))) = 24 (1 * (3 * (2 * 4))) = 24 (1 * (3 * (4 * 2))) = 24 (1 * (4 * (2 * 3))) = 24 (1 * (4 * (3 * 2))) = 24 (2 * (1 * (3 * 4))) = 24 (2 * (1 * (4 * 3))) = 24 (2 * (3 * (1 * 4))) = 24 (2 * (3 * (4 * 1))) = 24 (2 * (4 * (1 * 3))) = 24 (2 * (4 * (3 * 1))) = 24 (3 * (1 * (2 * 4))) = 24 (3 * (1 * (4 * 2))) = 24 (3 * (2 * (1 * 4))) = 24 (3 * (2 * (4 * 1))) = 24 (3 * (4 * (1 * 2))) = 24 (3 * (4 * (2 * 1))) = 24 (4 * (1 * (2 * 3))) = 24 (4 * (1 * (3 * 2))) = 24 (4 * (2 * (1 * 3))) = 24 (4 * (2 * (3 * 1))) = 24 (4 * (3 * (1 * 2))) = 24 (4 * (3 * (2 * 1))) = 24 (1 * ((2 * 3) * 4)) = 24 (1 * ((2 * 4) * 3)) = 24 (1 * ((3 * 2) * 4)) = 24 (1 * ((3 * 4) * 2)) = 24 (1 * ((4 * 2) * 3)) = 24 (1 * ((4 * 3) * 2)) = 24 (2 * ((1 * 3) * 4)) = 24 (2 * ((1 * 4) * 3)) = 24 (2 * ((3 * 1) * 4)) = 24 (2 * ((3 * 4) * 1)) = 24 (2 * ((4 * 1) * 3)) = 24 (2 * ((4 * 3) * 1)) = 24 (3 * ((1 * 2) * 4)) = 24 (3 * ((1 * 4) * 2)) = 24 (3 * ((2 * 1) * 4)) = 24 (3 * ((2 * 4) * 1)) = 24 (3 * ((4 * 1) * 2)) = 24 (3 * ((4 * 2) * 1)) = 24 (4 * ((1 * 2) * 3)) = 24 (4 * ((1 * 3) * 2)) = 24 (4 * ((2 * 1) * 3)) = 24 (4 * ((2 * 3) * 1)) = 24 (4 * ((3 * 1) * 2)) = 24 (4 * ((3 * 2) * 1)) = 24 ((1 * 2) * (3 * 4)) = 24 ((1 * 2) * (4 * 3)) = 24 ((1 * 3) * (2 * 4)) = 24 ((1 * 3) * (4 * 2)) = 24 ((1 * 4) * (2 * 3)) = 24 ((1 * 4) * (3 * 2)) = 24 ((2 * 1) * (3 * 4)) = 24 ((2 * 1) * (4 * 3)) = 24 ((2 * 3) * (1 * 4)) = 24 ((2 * 3) * (4 * 1)) = 24 ((2 * 4) * (1 * 3)) = 24 ((2 * 4) * (3 * 1)) = 24 ((3 * 1) * (2 * 4)) = 24 ((3 * 1) * (4 * 2)) = 24 ((3 * 2) * (1 * 4)) = 24 ((3 * 2) * (4 * 1)) = 24 ((3 * 4) * (1 * 2)) = 24 ((3 * 4) * (2 * 1)) = 24 ((4 * 1) * (2 * 3)) = 24 ((4 * 1) * (3 * 2)) = 24 ((4 * 2) * (1 * 3)) = 24 ((4 * 2) * (3 * 1)) = 24 ((4 * 3) * (1 * 2)) = 24 ((4 * 3) * (2 * 1)) = 24 ((1 * (2 * 3)) * 4) = 24 ((1 * (2 * 4)) * 3) = 24 ((1 * (3 * 2)) * 4) = 24 ((1 * (3 * 4)) * 2) = 24 ((1 * (4 * 2)) * 3) = 24 ((1 * (4 * 3)) * 2) = 24 ((2 * (1 * 3)) * 4) = 24 ((2 * (1 * 4)) * 3) = 24 ((2 * (3 * 1)) * 4) = 24 ((2 * (3 * 4)) * 1) = 24 ((2 * (4 * 1)) * 3) = 24 ((2 * (4 * 3)) * 1) = 24 ((3 * (1 * 2)) * 4) = 24 ((3 * (1 * 4)) * 2) = 24 ((3 * (2 * 1)) * 4) = 24 ((3 * (2 * 4)) * 1) = 24 ((3 * (4 * 1)) * 2) = 24 ((3 * (4 * 2)) * 1) = 24 ((4 * (1 * 2)) * 3) = 24 ((4 * (1 * 3)) * 2) = 24 ((4 * (2 * 1)) * 3) = 24 ((4 * (2 * 3)) * 1) = 24 ((4 * (3 * 1)) * 2) = 24 ((4 * (3 * 2)) * 1) = 24 (((1 * 2) * 3) * 4) = 24 (((1 * 2) * 4) * 3) = 24 (((1 * 3) * 2) * 4) = 24 (((1 * 3) * 4) * 2) = 24 (((1 * 4) * 2) * 3) = 24 (((1 * 4) * 3) * 2) = 24 (((2 * 1) * 3) * 4) = 24 (((2 * 1) * 4) * 3) = 24 (((2 * 3) * 1) * 4) = 24 (((2 * 3) * 4) * 1) = 24 (((2 * 4) * 1) * 3) = 24 (((2 * 4) * 3) * 1) = 24 (((3 * 1) * 2) * 4) = 24 (((3 * 1) * 4) * 2) = 24 (((3 * 2) * 1) * 4) = 24 (((3 * 2) * 4) * 1) = 24 (((3 * 4) * 1) * 2) = 24 (((3 * 4) * 2) * 1) = 24 (((4 * 1) * 2) * 3) = 24 (((4 * 1) * 3) * 2) = 24 (((4 * 2) * 1) * 3) = 24 (((4 * 2) * 3) * 1) = 24 (((4 * 3) * 1) * 2) = 24 (((4 * 3) * 2) * 1) = 24 ((2 * (3 * 4)) / 1) = 24 ((2 * (4 * 3)) / 1) = 24 ((3 * (2 * 4)) / 1) = 24 ((3 * (4 * 2)) / 1) = 24 ((4 * (2 * 3)) / 1) = 24 ((4 * (3 * 2)) / 1) = 24 (((2 * 3) * 4) / 1) = 24 (((2 * 4) * 3) / 1) = 24 (((3 * 2) * 4) / 1) = 24 (((3 * 4) * 2) / 1) = 24 (((4 * 2) * 3) / 1) = 24 (((4 * 3) * 2) / 1) = 24 (2 * ((3 * 4) / 1)) = 24 (2 * ((4 * 3) / 1)) = 24 (3 * ((2 * 4) / 1)) = 24 (3 * ((4 * 2) / 1)) = 24 (4 * ((2 * 3) / 1)) = 24 (4 * ((3 * 2) / 1)) = 24 ((2 * 3) * (4 / 1)) = 24 ((2 * 4) * (3 / 1)) = 24 ((3 * 2) * (4 / 1)) = 24 ((3 * 4) * (2 / 1)) = 24 ((4 * 2) * (3 / 1)) = 24 ((4 * 3) * (2 / 1)) = 24 (((2 * 3) / 1) * 4) = 24 (((2 * 4) / 1) * 3) = 24 (((3 * 2) / 1) * 4) = 24 (((3 * 4) / 1) * 2) = 24 (((4 * 2) / 1) * 3) = 24 (((4 * 3) / 1) * 2) = 24 (2 / (1 / (3 * 4))) = 24 (2 / (1 / (4 * 3))) = 24 (3 / (1 / (2 * 4))) = 24 (3 / (1 / (4 * 2))) = 24 (4 / (1 / (2 * 3))) = 24 (4 / (1 / (3 * 2))) = 24 ((2 * 3) / (1 / 4)) = 24 ((2 * 4) / (1 / 3)) = 24 ((3 * 2) / (1 / 4)) = 24 ((3 * 4) / (1 / 2)) = 24 ((4 * 2) / (1 / 3)) = 24 ((4 * 3) / (1 / 2)) = 24 (2 * (3 * (4 / 1))) = 24 (2 * (4 * (3 / 1))) = 24 (3 * (2 * (4 / 1))) = 24 (3 * (4 * (2 / 1))) = 24 (4 * (2 * (3 / 1))) = 24 (4 * (3 * (2 / 1))) = 24 (2 * ((3 / 1) * 4)) = 24 (2 * ((4 / 1) * 3)) = 24 (3 * ((2 / 1) * 4)) = 24 (3 * ((4 / 1) * 2)) = 24 (4 * ((2 / 1) * 3)) = 24 (4 * ((3 / 1) * 2)) = 24 ((2 / 1) * (3 * 4)) = 24 ((2 / 1) * (4 * 3)) = 24 ((3 / 1) * (2 * 4)) = 24 ((3 / 1) * (4 * 2)) = 24 ((4 / 1) * (2 * 3)) = 24 ((4 / 1) * (3 * 2)) = 24 ((2 * (3 / 1)) * 4) = 24 ((2 * (4 / 1)) * 3) = 24 ((3 * (2 / 1)) * 4) = 24 ((3 * (4 / 1)) * 2) = 24 ((4 * (2 / 1)) * 3) = 24 ((4 * (3 / 1)) * 2) = 24 (((2 / 1) * 3) * 4) = 24 (((2 / 1) * 4) * 3) = 24 (((3 / 1) * 2) * 4) = 24 (((3 / 1) * 4) * 2) = 24 (((4 / 1) * 2) * 3) = 24 (((4 / 1) * 3) * 2) = 24 (2 * (3 / (1 / 4))) = 24 (2 * (4 / (1 / 3))) = 24 (3 * (2 / (1 / 4))) = 24 (3 * (4 / (1 / 2))) = 24 (4 * (2 / (1 / 3))) = 24 (4 * (3 / (1 / 2))) = 24 ((2 / (1 / 3)) * 4) = 24 ((2 / (1 / 4)) * 3) = 24 ((3 / (1 / 2)) * 4) = 24 ((3 / (1 / 4)) * 2) = 24 ((4 / (1 / 2)) * 3) = 24 ((4 / (1 / 3)) * 2) = 24 (2 / ((1 / 3) / 4)) = 24 (2 / ((1 / 4) / 3)) = 24 (3 / ((1 / 2) / 4)) = 24 (3 / ((1 / 4) / 2)) = 24 (4 / ((1 / 2) / 3)) = 24 (4 / ((1 / 3) / 2)) = 24
对于一些平时口算相对稍难的一些组合也是毫无压力，例如{1, 5, 5, 5}， {3, 3, 7, 7}， {3, 3, 8, 8}，有兴趣的看官口算试试 :)