如何找到的操作的确切设定为特定号码？ [英] How to find the exact set of operations for a specific number?
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问题描述
我想实现一个程序，回答这个问题：
I am trying to implement a program that answers this problem :
如果给你一个具体的数字（例如：268）
和其他6个号码（例如：2,4,5,25,75,100）
if you are given a specific number (for example : 268)
and another 6 numbers (for example : 2,4,5,25,75,100)
我怎样才能找到给我确切的答案或最接近的一种，它的操作？
How can I find the operation that gives me the exact answer or the closest one to it?
您可以通过使用该操作接听previous例如：75 * 4-25-5-2 = 268
You can answer the previous example by using this operation : 75*4-25-5-2 = 268
规则：
- 您可以使用这些算术运算：+， - ，*，/，（）
- 当你用除法的提醒必须等于0（6/3是确定的，但 6/4也不行！）。
- 您不能使用超过一次相同数量更多。 此外，您可以尽量避免使用数字（例如：100我们 previous例子）。
- you can use these arithmetic operations : +, -, *, / , ().
- when you use division the reminder must be equal to 0 (6/3 is ok but 6/4 is not ok!).
- you can not use the same number more than once. Also, you can avoid using a number (for example : 100 in our previous example).
时有什么比写全的可能性，并采取最近的一个更好的解决办法？因为这个答案会强迫我写的code过多的线条，非常感谢！
Is there is any better solution than writing the whole possibilities and taking the closest one? Because this answer would force me to write too many lines of code, thanks!
推荐答案
事实上，蛮力解决方案确实是没有那么多code
In fact, the brute force solution really isn't so much code
（编辑：改变了code略有下降，因为一些规则没有得到适当的考虑）
( Changed the code slightly, because some of the rules had not been properly taken into account)
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
public class NumberPuzzle
{
public static void main(String[] args)
{
List<Integer> numbers = Arrays.asList(2,4,5,25,75,100);
Integer result = 268;
solve(numbers, result);
}
private static void solve(List<Integer> numbers, Integer result)
{
List<Node> nodes = new ArrayList<Node>();
for (int i=0; i<numbers.size(); i++)
{
Integer number = numbers.get(i);
nodes.add(new Node(number));
}
System.out.println(nodes);
List<Node> all = create(nodes);
System.out.println("Found "+all.size()+" combinations");
List<Node> best = new ArrayList<Node>();
Integer minDifference = Integer.MAX_VALUE;
for (Node n : all)
{
//System.out.println(n);
Integer e = n.evaluate();
Integer difference = Math.abs(e - result);
if (difference < minDifference)
{
best.clear();
minDifference = difference;
best.add(n);
}
else if (difference.equals(minDifference))
{
best.add(n);
}
}
for (Node n : best)
{
System.out.println(n+" = "+n.evaluate());
}
}
private static List<Node> create(List<Node> nodes)
{
if (nodes.size() == 1)
{
return nodes;
}
List<Node> result = new ArrayList<Node>(nodes);
for (int i=0; i<nodes.size(); i++)
{
List<Node> copy = new ArrayList<Node>(nodes);
Node node = copy.remove(i);
List<Node> others = create(copy);
for (int j=0; j<others.size(); j++)
{
Node other = others.get(j);
result.add(new Node(node, '+', other));
result.add(new Node(node, '*', other));
result.add(new Node(node, '-', other));
result.add(new Node(other, '-', node));
Integer vNode = node.evaluate();
Integer vOther = other.evaluate();
if (vOther != 0 && vNode % vOther == 0)
{
result.add(new Node(node, '/', other));
}
if (vNode != 0 && vOther % vNode == 0)
{
result.add(new Node(other, '/', node));
}
}
}
return result;
}
static class Node
{
Integer value;
Node left;
Character op;
Node right;
Node(Node left, Character op, Node right)
{
this.left = left;
this.op = op;
this.right = right;
}
Node(Integer value)
{
this.value = value;
}
Integer evaluate()
{
if (op != null)
{
Integer lv = left.evaluate();
Integer rv = right.evaluate();
switch (op)
{
case '+': return lv + rv;
case '-': return lv - rv;
case '*': return lv * rv;
case '/': return rv.equals(0) ? Integer.MAX_VALUE : lv / rv;
}
}
return value;
}
@Override
public String toString()
{
if (op == null)
{
return String.valueOf(value);
}
return "("+left.toString()+op+right.toString()+")";
}
}
}
据发现颇有些解决方案......
It finds quite some solutions....
（编辑：根据变更后的code更新）
( Updated according to the changed code)
(2*(4+(5+(25+100)))) = 268
(2*(4+(5+(100+25)))) = 268
(2*(4+(25+(5+100)))) = 268
(2*(4+(25+(100+5)))) = 268
(2*(4+(100+(5+25)))) = 268
(2*(4+(100+(25+5)))) = 268
((5*(4-(25-75)))-2) = 268
((5*(4+(75-25)))-2) = 268
(2*(5+(4+(25+100)))) = 268
((5*(4+(25-(75-100))))-2) = 268
((5*(4-((75-100)-25)))-2) = 268
((5*(4+(25+(100-75))))-2) = 268
((5*(4+(25+(100-75))))-2) = 268
((5*(4+(25-(75-100))))-2) = 268
((5*(4-((75-100)-25)))-2) = 268
((5*(4+(75-25)))-2) = 268
((5*(4-(25-75)))-2) = 268
((5*(4-(75-(25+100))))-2) = 268
((5*(4+((25+100)-75)))-2) = 268
((5*(4-(75-(100+25))))-2) = 268
((5*(4+((100+25)-75)))-2) = 268
(2*(5+(4+(100+25)))) = 268
((5*(4+(100+(25-75))))-2) = 268
((5*(4+(100-(75-25))))-2) = 268
((5*(4-((75-25)-100)))-2) = 268
((5*(4+(100-(75-25))))-2) = 268
((5*(4-((75-25)-100)))-2) = 268
((5*(4+(100+(25-75))))-2) = 268
((5*((4+75)-25))-2) = 268
((((4*75)-25)-5)-2) = 268
(2*(5+(25+(4+100)))) = 268
((5*(25+(4-(75-100))))-2) = 268
((5*(25-((75-100)-4)))-2) = 268
((5*(25+(4+(100-75))))-2) = 268
((5*(25+(4+(100-75))))-2) = 268
((5*(25+(4-(75-100))))-2) = 268
((5*(25-((75-100)-4)))-2) = 268
((5*((75+4)-25))-2) = 268
((((75*4)-25)-5)-2) = 268
((5*(25-(75-(4+100))))-2) = 268
((5*(25+((4+100)-75)))-2) = 268
((5*(25-(75-(100+4))))-2) = 268
((5*(25+((100+4)-75)))-2) = 268
(2*(5+(25+(100+4)))) = 268
((5*(25+(100+(4-75))))-2) = 268
((5*(25+(100-(75-4))))-2) = 268
((5*(25-((75-4)-100)))-2) = 268
((5*(25+(100-(75-4))))-2) = 268
((5*(25-((75-4)-100)))-2) = 268
((5*(25+(100+(4-75))))-2) = 268
((5*(75+(4-25)))-2) = 268
((5*(75-(25-4)))-2) = 268
((5*((4+(25+100))-75))-2) = 268
((5*((4+(100+25))-75))-2) = 268
((5*(75-(25-4)))-2) = 268
((5*(75+(4-25)))-2) = 268
((5*((25+(4+100))-75))-2) = 268
((5*((25+(100+4))-75))-2) = 268
((5*((100+(4+25))-75))-2) = 268
(((75+(100+(4*25)))-5)-2) = 268
((5*((100+(25+4))-75))-2) = 268
(((75+(100+(25*4)))-5)-2) = 268
(2*(5+(100+(4+25)))) = 268
((5*(100+(4+(25-75))))-2) = 268
((5*(100+(4-(75-25))))-2) = 268
((5*(100-((75-25)-4)))-2) = 268
((5*(100+(4-(75-25))))-2) = 268
((5*(100-((75-25)-4)))-2) = 268
((5*(100+(4+(25-75))))-2) = 268
(2*(5+(100+(25+4)))) = 268
((5*(100+(25+(4-75))))-2) = 268
((5*(100+(25-(75-4))))-2) = 268
((5*(100-((75-4)-25)))-2) = 268
((5*(100+(25-(75-4))))-2) = 268
((5*(100-((75-4)-25)))-2) = 268
((5*(100+(25+(4-75))))-2) = 268
((5*(100-(75-(4+25))))-2) = 268
((5*(100+((4+25)-75)))-2) = 268
(((100+(75+(4*25)))-5)-2) = 268
((5*(100-(75-(25+4))))-2) = 268
((5*(100+((25+4)-75)))-2) = 268
(((100+(75+(25*4)))-5)-2) = 268
(2*(25+(4+(5+100)))) = 268
(2*(25+(4+(100+5)))) = 268
((((4*75)-5)-25)-2) = 268
(2*(25+(5+(4+100)))) = 268
((((75*4)-5)-25)-2) = 268
(2*(25+(5+(100+4)))) = 268
(2*(25+(100+(4+5)))) = 268
(2*(25+(100+(5+4)))) = 268
((((5*(4+75))-100)-25)-2) = 268
((((5*(75+4))-100)-25)-2) = 268
((75-(5-(100+(4*25))))-2) = 268
((75+((100+(4*25))-5))-2) = 268
((75-(5-(100+(25*4))))-2) = 268
((75+((100+(25*4))-5))-2) = 268
((75+(100-(5-(4*25))))-2) = 268
((75-((5-(4*25))-100))-2) = 268
((75+(100+((4*25)-5)))-2) = 268
((75+(100-(5-(25*4))))-2) = 268
((75-((5-(25*4))-100))-2) = 268
((75+(100+((25*4)-5)))-2) = 268
(2*(100+(4+(5+25)))) = 268
(2*(100+(4+(25+5)))) = 268
(2*(100+(5+(4+25)))) = 268
(2*(100+(5+(25+4)))) = 268
((100-(5-(75+(4*25))))-2) = 268
((100+((75+(4*25))-5))-2) = 268
((100-(5-(75+(25*4))))-2) = 268
((100+((75+(25*4))-5))-2) = 268
(2*(100+(25+(4+5)))) = 268
(2*(100+(25+(5+4)))) = 268
((((5*(4+75))-25)-100)-2) = 268
((((5*(75+4))-25)-100)-2) = 268
((100+(75-(5-(4*25))))-2) = 268
((100-((5-(4*25))-75))-2) = 268
((100+(75+((4*25)-5)))-2) = 268
((100+(75-(5-(25*4))))-2) = 268
((100-((5-(25*4))-75))-2) = 268
((100+(75+((25*4)-5)))-2) = 268
(((100*((75-5)-2))/25)-4) = 268
(((100*((75-5)-2))/25)-4) = 268
(((100*((75-2)-5))/25)-4) = 268
(((100*((75-2)-5))/25)-4) = 268
(((100*(75-(2+5)))/25)-4) = 268
(((100*(75-(5+2)))/25)-4) = 268
(4*(75-(2*(100/25)))) = 268
(4*(75-(2*(100/25)))) = 268
(4*(75-((2*100)/25))) = 268
(4*(75-((100*2)/25))) = 268
((((4*75)-25)-2)-5) = 268
((((75*4)-25)-2)-5) = 268
(((75+(100+(4*25)))-2)-5) = 268
(((75+(100+(25*4)))-2)-5) = 268
(((100+(75+(4*25)))-2)-5) = 268
(((100+(75+(25*4)))-2)-5) = 268
((((4*75)-2)-25)-5) = 268
((((75*4)-2)-25)-5) = 268
((75-(2-(100+(4*25))))-5) = 268
((75+((100+(4*25))-2))-5) = 268
((75-(2-(100+(25*4))))-5) = 268
((75+((100+(25*4))-2))-5) = 268
((75+(100-(2-(4*25))))-5) = 268
((75-((2-(4*25))-100))-5) = 268
((75+(100+((4*25)-2)))-5) = 268
((75+(100-(2-(25*4))))-5) = 268
((75-((2-(25*4))-100))-5) = 268
((75+(100+((25*4)-2)))-5) = 268
((100-(2-(75+(4*25))))-5) = 268
((100+((75+(4*25))-2))-5) = 268
((100-(2-(75+(25*4))))-5) = 268
((100+((75+(25*4))-2))-5) = 268
((100+(75-(2-(4*25))))-5) = 268
((100-((2-(4*25))-75))-5) = 268
((100+(75+((4*25)-2)))-5) = 268
((100+(75-(2-(25*4))))-5) = 268
((100-((2-(25*4))-75))-5) = 268
((100+(75+((25*4)-2)))-5) = 268
((((4*75)-5)-2)-25) = 268
((((75*4)-5)-2)-25) = 268
((((5*(4+75))-100)-2)-25) = 268
((((5*(75+4))-100)-2)-25) = 268
((((4*75)-2)-5)-25) = 268
((((75*4)-2)-5)-25) = 268
((75+(2*(4+(5+100))))-25) = 268
((75+(2*(4+(100+5))))-25) = 268
((75+(2*(5+(4+100))))-25) = 268
((75+(2*(5+(100+4))))-25) = 268
((75+(2*(100+(4+5))))-25) = 268
((75+(2*(100+(5+4))))-25) = 268
((((5*(4+75))-2)-100)-25) = 268
((((5*(75+4))-2)-100)-25) = 268
((100*(75-(2*4)))/25) = 268
((100*(75-(4*2)))/25) = 268
(75-(2+(5-(100+(4*25))))) = 268
(75-(2-((100+(4*25))-5))) = 268
(75+(((100+(4*25))-5)-2)) = 268
(75-(2+(5-(100+(25*4))))) = 268
(75-(2-((100+(25*4))-5))) = 268
(75+(((100+(25*4))-5)-2)) = 268
(75-(2-(100-(5-(4*25))))) = 268
(75+((100-(5-(4*25)))-2)) = 268
(75-(2+((5-(4*25))-100))) = 268
(75-(2-(100+((4*25)-5)))) = 268
(75+((100+((4*25)-5))-2)) = 268
(75-(2-(100-(5-(25*4))))) = 268
(75+((100-(5-(25*4)))-2)) = 268
(75-(2+((5-(25*4))-100))) = 268
(75-(2-(100+((25*4)-5)))) = 268
(75+((100+((25*4)-5))-2)) = 268
(75-(5+(2-(100+(4*25))))) = 268
(75-(5-((100+(4*25))-2))) = 268
(75+(((100+(4*25))-2)-5)) = 268
(75-(5+(2-(100+(25*4))))) = 268
(75-(5-((100+(25*4))-2))) = 268
(75+(((100+(25*4))-2)-5)) = 268
(75-(5-(100-(2-(4*25))))) = 268
(75+((100-(2-(4*25)))-5)) = 268
(75-(5+((2-(4*25))-100))) = 268
(75-(5-(100+((4*25)-2)))) = 268
(75+((100+((4*25)-2))-5)) = 268
(75-(5-(100-(2-(25*4))))) = 268
(75+((100-(2-(25*4)))-5)) = 268
(75-(5+((2-(25*4))-100))) = 268
(75-(5-(100+((25*4)-2)))) = 268
(75+((100+((25*4)-2))-5)) = 268
(75-(25-(2*(4+(5+100))))) = 268
(75+((2*(4+(5+100)))-25)) = 268
(75-(25-(2*(4+(100+5))))) = 268
(75+((2*(4+(100+5)))-25)) = 268
(75-(25-(2*(5+(4+100))))) = 268
(75+((2*(5+(4+100)))-25)) = 268
(75-(25-(2*(5+(100+4))))) = 268
(75+((2*(5+(100+4)))-25)) = 268
(75-(25-(2*(100+(4+5))))) = 268
(75+((2*(100+(4+5)))-25)) = 268
(75-(25-(2*(100+(5+4))))) = 268
(75+((2*(100+(5+4)))-25)) = 268
(75+(100-(2+(5-(4*25))))) = 268
(75-((2+(5-(4*25)))-100)) = 268
(75+(100-(2-((4*25)-5)))) = 268
(75-((2-((4*25)-5))-100)) = 268
(75+(100+(((4*25)-5)-2))) = 268
(75+(100-(2+(5-(25*4))))) = 268
(75-((2+(5-(25*4)))-100)) = 268
(75+(100-(2-((25*4)-5)))) = 268
(75-((2-((25*4)-5))-100)) = 268
(75+(100+(((25*4)-5)-2))) = 268
(75+(100-(5+(2-(4*25))))) = 268
(75-((5+(2-(4*25)))-100)) = 268
(75+(100-(5-((4*25)-2)))) = 268
(75-((5-((4*25)-2))-100)) = 268
(75+(100+(((4*25)-2)-5))) = 268
(75+(100-(5+(2-(25*4))))) = 268
(75-((5+(2-(25*4)))-100)) = 268
(75+(100-(5-((25*4)-2)))) = 268
(75-((5-((25*4)-2))-100)) = 268
(75+(100+(((25*4)-2)-5))) = 268
(100+(2*(4+(5+75)))) = 268
(100+(2*(4+(75+5)))) = 268
(100+(2*(4+(75+(25/5))))) = 268
(100+(2*(4+(75+(25/5))))) = 268
(100+(2*(5+(4+75)))) = 268
(100+(2*(5+(75+4)))) = 268
(100-(2+(5-(75+(4*25))))) = 268
(100-(2-((75+(4*25))-5))) = 268
(100+(((75+(4*25))-5)-2)) = 268
(100-(2+(5-(75+(25*4))))) = 268
(100-(2-((75+(25*4))-5))) = 268
(100+(((75+(25*4))-5)-2)) = 268
((((5*(4+75))-25)-2)-100) = 268
((((5*(75+4))-25)-2)-100) = 268
(100+(2*(75+(4+5)))) = 268
(100+(2*(75+(4+(25/5))))) = 268
(100+(2*(75+(4+(25/5))))) = 268
(100+(2*(75+(5+4)))) = 268
(100-(2-(75-(5-(4*25))))) = 268
(100+((75-(5-(4*25)))-2)) = 268
(100-(2+((5-(4*25))-75))) = 268
(100-(2-(75+((4*25)-5)))) = 268
(100+((75+((4*25)-5))-2)) = 268
(100-(2-(75-(5-(25*4))))) = 268
(100+((75-(5-(25*4)))-2)) = 268
(100-(2+((5-(25*4))-75))) = 268
(100-(2-(75+((25*4)-5)))) = 268
(100+((75+((25*4)-5))-2)) = 268
(100+(4*(2+(25+(75/5))))) = 268
(100+(4*(2+(25+(75/5))))) = 268
(100+(4*(25+(2+(75/5))))) = 268
(100+(4*(25+(2+(75/5))))) = 268
(100-(5+(2-(75+(4*25))))) = 268
(100-(5-((75+(4*25))-2))) = 268
(100+(((75+(4*25))-2)-5)) = 268
(100-(5+(2-(75+(25*4))))) = 268
(100-(5-((75+(25*4))-2))) = 268
(100+(((75+(25*4))-2)-5)) = 268
(100-(5-(75-(2-(4*25))))) = 268
(100+((75-(2-(4*25)))-5)) = 268
(100-(5+((2-(4*25))-75))) = 268
(100-(5-(75+((4*25)-2)))) = 268
(100+((75+((4*25)-2))-5)) = 268
(100-(5-(75-(2-(25*4))))) = 268
(100+((75-(2-(25*4)))-5)) = 268
(100-(5+((2-(25*4))-75))) = 268
(100-(5-(75+((25*4)-2)))) = 268
(100+((75+((25*4)-2))-5)) = 268
((((5*(4+75))-2)-25)-100) = 268
((((5*(75+4))-2)-25)-100) = 268
(100+(75-(2+(5-(4*25))))) = 268
(100-((2+(5-(4*25)))-75)) = 268
(100+(75-(2-((4*25)-5)))) = 268
(100-((2-((4*25)-5))-75)) = 268
(100+(75+(((4*25)-5)-2))) = 268
(100+(75-(2+(5-(25*4))))) = 268
(100-((2+(5-(25*4)))-75)) = 268
(100+(75-(2-((25*4)-5)))) = 268
(100-((2-((25*4)-5))-75)) = 268
(100+(75+(((25*4)-5)-2))) = 268
(100+(75-(5+(2-(4*25))))) = 268
(100-((5+(2-(4*25)))-75)) = 268
(100+(75-(5-((4*25)-2)))) = 268
(100-((5-((4*25)-2))-75)) = 268
(100+(75+(((4*25)-2)-5))) = 268
(100+(75-(5+(2-(25*4))))) = 268
(100-((5+(2-(25*4)))-75)) = 268
(100+(75-(5-((25*4)-2)))) = 268
(100-((5-((25*4)-2))-75)) = 268
(100+(75+(((25*4)-2)-5))) = 268
，但当然，许多这些都是由于可交换多余。
but of course, MANY of these are redundant due to commutativity.
人们可以很容易地避免许多是在强力变种检查了解决方案。其中的第一个步骤可能是避免建立包含交换元素的节点，并有可能实施等于方法快速和肮脏的节点类，我勾勒出现，这需要交换性考虑进去，然后用设置取值节点，而不是列表。而进一步，简单，低层次的优化是可能的，以及（在一个完美的匹配找到如早日回归）。
One could easily avoid many of the solutions that are checked in the brute-force variant. One of the first steps could be to avoid the creation of nodes that contain commutative elements, and possibly implement an equals method for the quick-and-dirty Node class that I sketched there, which takes commutativity into account, and then use Sets of Nodes instead of the List. And further, "simple", "low level" optimizations may be possible as well (e.g. an early return when a perfect match is found).
和涉及您的实际问题，是否有解决方案，更好比蛮力：我的直觉，感觉是，答案是不，但是这取决于该问题应该得到解决。关键的一点是在这里：假设你给出可用的号码的列表，以及所希望的结果值。并假定，这是不可能创建从给定的输入值的结果值。例如。当你有
And concerning your actual question, whether there is a solution that is "better" than brute force: My gut-feeling is that the answer is "no", but this depends on which problem should be solved. The key point is here: Assume that you are given a list of available numbers, and a desired result value. And assume, that it is not possible to create the result value from the given input values. E.g. when you have
input = { 1,2,3,4,5,6 }
result = 10000000000;
我觉得要想真正的证明，这是不可能的，您有无枚举所有组合（返回最好一到底，在这里可能是 1 * 2 * 3 * 4 * 5 * 6 ）。如果跳过任何的组合，你如何确保它是不完全的最佳的呢？
I think that in order to really prove that this is not possible, you have to enumerate all combinations (returning the "best" one in the end, which here will probably be 1*2*3*4*5*6). If you skip any combination, how should you be sure that it was not exactly the best one?
但同样：这只是一种直觉。也许有人谁更...数学参与......可以证明我错了......
But again: This is just a gut feeling. Maybe someone who is more ... mathematically involved ... can prove me wrong ....
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