464. Can I Win

In the "100 game," two players take turns adding, to a running total, any integer from 1..10. The player who first causes the running total to reach or exceed 100 wins.

What if we change the game so that players cannot re-use integers?

For example, two players might take turns drawing from a common pool of numbers of 1..15 without replacement until they reach a total >= 100.

Given an integer maxChoosableInteger and another integer desiredTotal, determine if the first player to move can force a win, assuming both players play optimally.

You can always assume that maxChoosableInteger will not be larger than 20 and desiredTotal will not be larger than 300.

Example

Input:
maxChoosableInteger = 10
desiredTotal = 11

Output:
false

Explanation:
No matter which integer the first player choose, the first player will lose.
The first player can choose an integer from 1 up to 10.
If the first player choose 1, the second player can only choose integers from 2 up to 10.
The second player will win by choosing 10 and get a total = 11, which is >= desiredTotal.
Same with other integers chosen by the first player, the second player will always win.
题目含义：两个人玩游戏，从1～maxChoosableInteger中任选一个数字，第一个人先选，第二个人后选，每个人选过的数字就不能再选了～两个人谁先加起来总和超过desiredTotal谁就赢，问给出数字，第一个人能否赢得比赛

    private boolean canWin(int[] flag,int desiredTotal,HashMap<String,Boolean> map) {
        String state=Arrays.toString(flag);
        if(map.containsKey(state))
            return map.get(state);
        for(int i=1;i<flag.length;i++){
            if(flag[i]==0){
                flag[i]=1;
                //当前获胜情况有2种：当前所选取的元素大于还剩余的差；选取当前元素之后，接下来的不能获胜，即对方怎么都输，则自己肯定赢。
                if(i>=desiredTotal||!canWin(flag,desiredTotal-i,map)){
                    map.put(state,true);
                    flag[i]=0;
                    return true;
                }
                flag[i]=0;
            }
        }
        map.put(state,false);
        return false;
    }
    
    public boolean canIWin(int maxChoosableInteger, int desiredTotal) {
//        动态规划，利用一个数组存储哪些元素已经被选了，然后接下来的游戏便不能选那些已选的元素
        if(desiredTotal<=maxChoosableInteger)
            return true;
        if(desiredTotal>maxChoosableInteger*(maxChoosableInteger+1)/2)
            return false;
        int[] flag=new int[maxChoosableInteger+1];
        return canWin(flag,desiredTotal,new HashMap<String,Boolean>());        
    }