Can I Win

                                            
class Solution {
public:
    bool canIWin(int maxChoosableInteger, int desiredTotal) {
        if (maxChoosableInteger >= desiredTotal) return true;
        if (maxChoosableInteger * (maxChoosableInteger + 1) / 2 < desiredTotal) return false;
        
        unordered_map<int, bool> memo;
        return canWin(maxChoosableInteger, desiredTotal, 0, memo);
    }
    
    bool canWin(int maxChoosableInteger, int desiredTotal, int chosen, unordered_map<int, bool>& memo) {
        if (memo.find(chosen) != memo.end()) return memo[chosen];
        
        for (int i = 1; i <= maxChoosableInteger; ++i) {
            int mask = 1 << i;
            if ((chosen & mask) == 0) {
                if (i >= desiredTotal || !canWin(maxChoosableInteger, desiredTotal - i, chosen | mask, memo)) {
                    memo[chosen] = true;
                    return true;
                }
            }
        }
        
        memo[chosen] = false;
        return false;
    }
};

                                            
import java.util.*;

class Solution {
    public boolean canIWin(int maxChoosableInteger, int desiredTotal) {
        if (desiredTotal <= maxChoosableInteger) {
            return true;
        }
        
        if (desiredTotal > (maxChoosableInteger + 1) * maxChoosableInteger / 2) {
            return false;
        }
        
        Map<Integer, Boolean> memo = new HashMap<>();
        return canWin(maxChoosableInteger, desiredTotal, 0, memo);
    }

    private boolean canWin(int maxChoosableInteger, int desiredTotal, int state, Map<Integer, Boolean> memo) {
        if (desiredTotal <= 0) {
            return false;
        }
        if (memo.containsKey(state)) {
            return memo.get(state);
        }
        for (int i = 0; i < maxChoosableInteger; i++) {
            if ((state & (1 << i)) == 0) {
                if (!canWin(maxChoosableInteger, desiredTotal - i - 1, state | (1 << i), memo)) {
                    memo.put(state, true);
                    return true;
                }
            }
        }
        memo.put(state, false);
        return false;
    }
}

                                            
class Solution(object):
    def canIWin(self, maxChoosableInteger, desiredTotal):
        if desiredTotal <= maxChoosableInteger:
            return True
        if sum(range(1, maxChoosableInteger + 1)) < desiredTotal:
            return False
        
        memo = {}
        def canWin(choices, total):
            if choices[-1] >= total:
                return True
            key = tuple(choices)
            if key in memo:
                return memo[key]
            
            for i in range(len(choices)):
                if not canWin(choices[:i] + choices[i+1:], total - choices[i]):
                    memo[key] = True
                    return True
                
            memo[key] = False
            return False
        
        return canWin(range(1, maxChoosableInteger + 1), desiredTotal)

                                            
public class Solution {
    public bool CanIWin(int maxChoosableInteger, int desiredTotal) {
        if (maxChoosableInteger >= desiredTotal) 
            return true;
        if ((1 + maxChoosableInteger) * maxChoosableInteger / 2 < desiredTotal)
            return false;
        
        return CanWin(0, new Dictionary<int, bool>(), maxChoosableInteger, desiredTotal);
    }
    
    private bool CanWin(int state, Dictionary<int, bool> memo, int maxChoosableInteger, int desiredTotal) {
        if (memo.ContainsKey(state))
            return memo[state];
        
        for (int i = 1; i <= maxChoosableInteger; i++) {
            int curr = 1 << i;
            if ((curr & state) == 0) {
                if (i >= desiredTotal || !CanWin(state | curr, memo, maxChoosableInteger, desiredTotal - i)) {
                    memo[state] = true;
                    return true;
                }
            }
        }
        
        memo[state] = false;
        return false;
    }
}

                                            
/**
 * @param {number} maxChoosableInteger
 * @param {number} desiredTotal
 * @return {boolean}
 */
var canIWin = function(maxChoosableInteger, desiredTotal) {
    if (maxChoosableInteger >= desiredTotal) return true;
    if ((1 + maxChoosableInteger) * maxChoosableInteger / 2 < desiredTotal) return false;

    const memo = new Map();

    const dfs = (maxChoosableInteger, desiredTotal, state) => {
        if (desiredTotal <= 0) return false;
        if (memo.has(state)) return memo.get(state);

        for (let i = 1; i <= maxChoosableInteger; i++) {
            if ((state & (1 << i)) === 0) {
                if (!dfs(maxChoosableInteger, desiredTotal - i, state | (1 << i))) {
                    memo.set(state, true);
                    return true;
                }
            }
        }

        memo.set(state, false);
        return false;
    };

    return dfs(maxChoosableInteger, desiredTotal, 0);
};

                                            
function canIWin(maxChoosableInteger: number, desiredTotal: number): boolean {
    if (desiredTotal === 0) return true;
    if ((maxChoosableInteger * (maxChoosableInteger + 1)) / 2 < desiredTotal) return false;
    const memo: Map<string, boolean> = new Map();

    function canWin(maxChoosableInteger: number, desiredTotal: number, chosen: number, memo: Map<string, boolean>): boolean {
        if (desiredTotal <= 0) return false;
        const key = chosen.toString();
        if (memo.has(key)) return memo.get(key)!;

        for (let i = 1; i <= maxChoosableInteger; i++) {
            if ((chosen & (1 << i)) === 0) {
                if (!canWin(maxChoosableInteger, desiredTotal - i, chosen | (1 << i), memo)) {
                    memo.set(key, true);
                    return true;
                }
            }
        }
        memo.set(key, false);
        return false;
    }

    return canWin(maxChoosableInteger, desiredTotal, 0, memo);
};

                                            
class Solution {
    /**
     * @param Integer $maxChoosableInteger
     * @param Integer $desiredTotal
     * @return Boolean
     */
    function canIWin($maxChoosableInteger, $desiredTotal) {
        if ($desiredTotal <= $maxChoosableInteger) {
            return true;
        }
        if (($maxChoosableInteger * ($maxChoosableInteger + 1) / 2) < $desiredTotal) {
            return false;
        }
        
        return $this->canWin(0, $desiredTotal, $maxChoosableInteger, 0);
    }
    
    function canWin($chosen, $total, $max, $mask) {
        if (isset($this->memo[$mask])) {
            return $this->memo[$mask];
        }
        
        for ($i = 1; $i <= $max; $i++) {
            $curr = 1 << $i;
            if (($mask & $curr) === 0) {
                if ($total <= $i || !$this->canWin($i, $total - $i, $max, $mask | $curr)) {
                    $this->memo[$mask] = true;
                    return true;
                }
            }
        }
        $this->memo[$mask] = false;
        return false;
    }
    
    private $memo = [];
}

                                            
class Solution {
    func canIWin(_ maxChoosableInteger: Int, _ desiredTotal: Int) -> Bool {
        if maxChoosableInteger >= desiredTotal {
            return true
        }
        if (1 + maxChoosableInteger) * maxChoosableInteger / 2 < desiredTotal {
            return false
        }
        var memo = [Int: Bool]()
        func canWin(_ total: Int, _ visited: Int) -> Bool {
            if let result = memo[visited] {
                return result
            }
            for i in 0..<maxChoosableInteger {
                let curr = 1 << i
                if (curr & visited) == 0 {
                    if total <= i + 1 || !canWin(total - (i + 1), visited | curr) {
                        memo[visited] = true
                        return true
                    }
                }
            }
            memo[visited] = false
            return false
        }
        
        return canWin(desiredTotal, 0)
    }
}

                                            
class Solution {
    fun canIWin(maxChoosableInteger: Int, desiredTotal: Int): Boolean {
        if (maxChoosableInteger >= desiredTotal) return true
        if (maxChoosableInteger * (maxChoosableInteger + 1) / 2 < desiredTotal) return false
        val map = HashMap<Int, Boolean>()
        return canWin(maxChoosableInteger, desiredTotal, 0, map)
    }

    private fun canWin(maxChoosableInteger: Int, desiredTotal: Int, state: Int, map: HashMap<Int, Boolean>): Boolean {
        if (desiredTotal <= 0) return false
        if (map.containsKey(state)) return map[state]!!

        for (i in 0 until maxChoosableInteger) {
            val move = (1 shl i)
            if ((state and move) == 0 && !canWin(maxChoosableInteger, desiredTotal - (i + 1), state or move, map)) {
                map[state] = true
                return true
            }
        }

        map[state] = false
        return false
    }
}

                                            
class Solution {
  bool canIWin(int maxChoosableInteger, int desiredTotal) {
    if (maxChoosableInteger >= desiredTotal) return true;
    if (maxChoosableInteger * (maxChoosableInteger + 1) ~/ 2 < desiredTotal) return false;
    
    return canWin(maxChoosableInteger, desiredTotal, 0, {}, {});
  }
  
  bool canWin(int maxChoosableInteger, int desiredTotal, int state, Map<int, bool> dp, Map<int, bool> visited) {
    if (desiredTotal <= 0) return false;
    
    if (visited.containsKey(state)) {
      return dp[state]!;
    }
    
    for (int i = 0; i < maxChoosableInteger; i++) {
      int curr = 1 << i;
      if ((state & curr) == 0) {
        if (!canWin(maxChoosableInteger, desiredTotal - i - 1, state | curr, dp, visited)) {
          dp[state] = true;
          visited[state] = true;
          return true;
        }
      }
    }
    
    dp[state] = false;
    visited[state] = true;
    return false;
  }
}

                                            
func canIWin(maxChoosableInteger int, desiredTotal int) bool {
    if desiredTotal <= 0 {
        return true
    }
    if (maxChoosableInteger*(maxChoosableInteger+1))/2 < desiredTotal {
        return false
    }
    memo := make(map[int]bool)
    return helper(maxChoosableInteger, desiredTotal, 0, memo)
}

func helper(maxChoosableInteger int, desiredTotal int, state int, memo map[int]bool) bool {
    if val, ok := memo[state]; ok {
        return val
    }
    for i := 1; i <= maxChoosableInteger; i++ {
        curr := 1 << (i - 1)
        if curr&state == 0 {
            if i >= desiredTotal || !helper(maxChoosableInteger, desiredTotal-i, curr|state, memo) {
                memo[state] = true
                return true
            }
        }
    }
    memo[state] = false
    return false
}

                                            
object Solution {
    def canIWin(maxChoosableInteger: Int, desiredTotal: Int): Boolean = {
        if (maxChoosableInteger >= desiredTotal) return true
        if (maxChoosableInteger * (maxChoosableInteger + 1) / 2 < desiredTotal) return false
        
        val dp = scala.collection.mutable.Map[Int, Boolean]()
        def canWin(remaining: Int, visited: Int): Boolean = {
            if (dp.contains(visited)) return dp(visited)
            
            var result = false
            var i = maxChoosableInteger
            while (i >= 1 && !result) {
                val mask = 1 << i
                if ((mask & visited) == 0) {
                    if (i >= remaining || !canWin(remaining - i, mask | visited)) {
                        result = true
                    }
                }
                i -= 1
            }
            
            dp(visited) = result
            result
        }
        
        canWin(desiredTotal, 0)
    }
}

                                            
impl Solution {
    pub fn can_i_win(max_choosable_integer: i32, desired_total: i32) -> bool {
        if max_choosable_integer >= desired_total {
            return true;
        }
        if (1 + max_choosable_integer) * max_choosable_integer / 2 < desired_total {
            return false;
        }
        
        let mut seen = std::collections::HashMap::new();
        
        fn can_win(max_choosable_integer: i32, desired_total: i32, state: i32, seen: &mut std::collections::HashMap<i32, bool>) -> bool {
            if let Some(&result) = seen.get(&state) {
                return result;
            }
            
            for i in 0..max_choosable_integer {
                let current = 1 << i;
                if current & state == 0 {
                    if desired_total <= i + 1 || !can_win(max_choosable_integer, desired_total - i - 1, state | current, seen) {
                        seen.insert(state, true);
                        return true;
                    }
                }
            }
            
            seen.insert(state, false);
            false
        }
        
        can_win(max_choosable_integer, desired_total, 0, &mut seen)
    }
}

To solve this coding challenge, we need to understand the game's rules and how to implement a strategy to determine if the first player can force a win. The two players take turns picking numbers from a fixed pool of integers without replacement until they reach or exceed a specified total. The goal is to determine if the first player can guarantee a win assuming both players use the best possible strategy.

Explanation

Initial Checks:

If the
desiredTotal

is less than or equal to the
maxChoosableInteger

, then the first player can choose a number equal to or greater than the
desiredTotal

and win immediately.
If the sum of all numbers from 1 to
maxChoosableInteger

is less than the
desiredTotal

, it is impossible to reach the desired total, so the first player cannot win.

Recursive Strategy with Memorization:

We need a recursive function
canWin

that takes the current choices left and the remaining total required to win.
If a player can force a win by picking any number from the remaining choices, we store this result to avoid recalculating it (memorization).

Memoization:

Use a dictionary to store results of subproblems to optimize the recursive calls.
The key for memoization is the current state of remaining choices.

Recursive Function Logic:

Check if the highest available number meets or exceeds the required total; if so, the current player wins.
For each number in the remaining choices, simulate picking that number and call
canWin

for the opponent with the updated choices and total.
If the opponent cannot win given the new state, it means the current player can force a win from this state.

Here is how we can translate this approach into pseudocode:

                                            
# Function to determine if the first player can force a win
function canIWin(maxChoosableInteger, desiredTotal):
# If the max number is greater than or equal to the desired total, first player wins immediately
if desiredTotal <= maxChoosableInteger:
return True

# If the sum of all numbers [1, 2, ..., maxChoosableInteger] is less than the desired total, no one can win
if sum(range(1, maxChoosableInteger + 1)) < desiredTotal:
return False

# Initialize memoization dictionary
memo = {}

# Internal helper function to determine if current player can force a win
function canWin(choices, total):
# If the largest number in choices meets or exceeds the total, current player wins
if choices[-1] >= total:
return True

# Convert choices to a tuple to use as a memoization key
key = tuple(choices)

# Check if the result for this state is already computed
if key in memo:
return memo[key]

# Iterate through each choice
for i in range(len(choices)):
# Simulate picking the current choice and checking if the opponent loses
if not canWin(choices[:i] + choices[i+1:], total - choices[i]):
# If the opponent cannot win, save and return True
memo[key] = True
return True

# If no forced win is found, save and return False
memo[key] = False
return False

# Initial call with all choices from 1 to maxChoosableInteger and the desired total
return canWin(range(1, maxChoosableInteger + 1), desiredTotal)

Step-by-Step Explanation

Initial Checks:

Check if
desiredTotal

can be reached in one move by checking if it is less than or equal to
maxChoosableInteger

.
Check if the sum of all possible choices is less than
desiredTotal

.

Setting Up Memoization:

Create an empty memoization dictionary to store results of subproblems.

Helper Function
canWin

:

Base case: If the largest number in the remaining choices is greater than or equal to the
total

, return
True

(indicates the current player can win).
Convert the list of remaining choices to a tuple to use as a key for memoization.
If this key is already in the memo dictionary, return the stored result.
Iterate through each number in the choices:

Simulate the current player picking that number and recursively call
canWin

to check if the opponent loses (if the opponent loses, it means the current player can win).
If the opponent cannot win from the new state, save
True

in memo and return
True

.

If no winning strategy is found for the current player, memoize and return
False

.

By following these steps, we ensure that we explore all possible combinations and make optimal decisions at each step to determine if the first player has a winning strategy. This approach uses dynamic programming principles to avoid redundant calculations, thus improving efficiency.