374. Guess Number Higher Or Lower - Explanation

Problem Link

Description

We are playing the Guess Game. The game is as follows:

I pick a number from 1 to n. You have to guess which number I picked.

Every time you guess wrong, I will tell you whether the number I picked is higher or lower than your guess.

You call a pre-defined API int guess(int num), which returns three possible results:

0: your guess is equal to the number I picked (i.e. num == pick).
-1: Your guess is higher than the number I picked (i.e. num > pick).
1: Your guess is lower than the number I picked (i.e. num < pick).

Return the number that I picked.

Example 1:

Input: n = 5, pick = 3

Output: 3

Example 2:

Input: n = 15, pick = 10

Output: 10

Example 3:

Input: n = 1, pick = 1

Output: 1

Constraints:

1 <= pick <= n <= ((2^31)-1)

Topics

Binary Search Interactive

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Prerequisites

Before attempting this problem, you should be comfortable with:

Binary Search - Efficiently searching a sorted range by repeatedly halving the search space based on comparison feedback

1. Linear Search

Intuition

The simplest approach is to try every number from 1 to n until we find the correct one. Each guess tells us whether we hit the target. While guaranteed to work, this method is slow for large n since we might need to check every single number.

Algorithm

Iterate through numbers from 1 to n.
For each number, call the guess API.
If guess returns 0, the current number is correct; return it.
Continue until the answer is found.

# The guess API is already defined for you.
# @param num, your guess
# @return -1 if num is higher than the picked number
#          1 if num is lower than the picked number
#          otherwise return 0
# def guess(num: int) -> int:

class Solution:
    def guessNumber(self, n: int) -> int:
        for num in range(1, n + 1):
            if guess(num) == 0:
                return num

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution extends GuessGame {
    public int guessNumber(int n) {
        for (int num = 1; num <= n; num++) {
            if (guess(num) == 0) return num;
        }
        return n;
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

class Solution {
public:
    int guessNumber(int n) {
        for (int num = 1; num <= n; num++) {
            if (guess(num) == 0) return num;
        }
        return n;
    }
};

/**
 * Forward declaration of guess API.
 * @param {number} num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * function guess(num) {}
 */

class Solution {
    /**
     * @param {number} n
     * @return {number}
     */
    guessNumber(n) {
        for (let num = 1; num <= n; num++) {
            if (guess(num) === 0) return num;
        }
        return n;
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution : GuessGame {
    public int GuessNumber(int n) {
        for (int num = 1; num <= n; num++) {
            if (guess(num) == 0) return num;
        }
        return n;
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(num int) int;
 */

func guessNumber(n int) int {
    for num := 1; num <= n; num++ {
        if guess(num) == 0 {
            return num
        }
    }
    return n
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * fun guess(num: Int): Int {}
 */

class Solution : GuessGame() {
    override fun guessNumber(n: Int): Int {
        for (num in 1..n) {
            if (guess(num) == 0) return num
        }
        return n
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(_ num: Int) -> Int
 */

class Solution : GuessGame {
    func guessNumber(_ n: Int) -> Int {
        for num in 1...n {
            if guess(num) == 0 {
                return num
            }
        }
        return n
    }
}

// Forward declaration of guess API.
// unsafe fn guess(num: i32) -> i32;

impl Solution {
    unsafe fn guessNumber(n: i32) -> i32 {
        for num in 1..=n {
            if guess(num) == 0 {
                return num;
            }
        }
        n
    }
}

Time & Space Complexity

Time complexity: $O(n)$
Space complexity: $O(1)$

2. Binary Search

Intuition

Since the numbers from 1 to n are sorted, we can use binary search to find the target efficiently. The guess API tells us whether to search higher or lower, which is exactly the feedback binary search needs to halve the search space with each guess.

Algorithm

Initialize two pointers: l = 1 and r = n.
Calculate the middle value m = (l + r) / 2.
Call guess(m):
- If it returns 0, we found the number; return m.
- If it returns 1 (number is higher), set l = m + 1.
- If it returns -1 (number is lower), set r = m - 1.
Repeat until the number is found.

# The guess API is already defined for you.
# @param num, your guess
# @return -1 if num is higher than the picked number
#          1 if num is lower than the picked number
#          otherwise return 0
# def guess(num: int) -> int:

class Solution:
    def guessNumber(self, n: int) -> int:
        l, r = 1, n
        while True:
            m = (l + r) // 2
            res = guess(m)
            if res > 0:
                l = m + 1
            elif res < 0:
                r = m - 1
            else:
                return m

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution extends GuessGame {
    public int guessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m = l + (r - l) / 2;
            int res = guess(m);
            if (res > 0) {
                l = m + 1;
            } else if (res < 0) {
                r = m - 1;
            } else {
                return m;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

class Solution {
public:
    int guessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m = l + (r - l) / 2;
            int res = guess(m);
            if (res > 0) {
                l = m + 1;
            } else if (res < 0) {
                r = m - 1;
            } else {
                return m;
            }
        }
    }
};

/**
 * Forward declaration of guess API.
 * @param {number} num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * function guess(num) {}
 */

class Solution {
    /**
     * @param {number} n
     * @return {number}
     */
    guessNumber(n) {
        let l = 1,
            r = n;
        while (true) {
            let m = Math.floor((l + r) / 2);
            let res = guess(m);
            if (res > 0) {
                l = m + 1;
            } else if (res < 0) {
                r = m - 1;
            } else {
                return m;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution : GuessGame {
    public int GuessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m = l + (r - l) / 2;
            int res = guess(m);
            if (res > 0) {
                l = m + 1;
            }
            else if (res < 0) {
                r = m - 1;
            }
            else {
                return m;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(num int) int;
 */

func guessNumber(n int) int {
    l, r := 1, n
    for {
        m := l + (r-l)/2
        res := guess(m)
        if res > 0 {
            l = m + 1
        } else if res < 0 {
            r = m - 1
        } else {
            return m
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * fun guess(num: Int): Int {}
 */

class Solution : GuessGame() {
    override fun guessNumber(n: Int): Int {
        var l = 1
        var r = n
        while (true) {
            val m = l + (r - l) / 2
            val res = guess(m)
            if (res > 0) {
                l = m + 1
            } else if (res < 0) {
                r = m - 1
            } else {
                return m
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(_ num: Int) -> Int
 */

class Solution : GuessGame {
    func guessNumber(_ n: Int) -> Int {
        var l = 1
        var r = n
        while true {
            let m = l + (r - l) / 2
            let res = guess(m)
            if res > 0 {
                l = m + 1
            } else if res < 0 {
                r = m - 1
            } else {
                return m
            }
        }
    }
}

// Forward declaration of guess API.
// unsafe fn guess(num: i32) -> i32;

impl Solution {
    unsafe fn guessNumber(n: i32) -> i32 {
        let (mut l, mut r) = (1, n);
        loop {
            let m = l + (r - l) / 2;
            let res = guess(m);
            if res > 0 {
                l = m + 1;
            } else if res < 0 {
                r = m - 1;
            } else {
                return m;
            }
        }
    }
}

Time & Space Complexity

Time complexity: $O(\log n)$
Space complexity: $O(1)$

3. Ternary Search

Intuition

Ternary search divides the search space into three parts instead of two. We pick two midpoints and use the guess API on both. Based on the results, we can eliminate either one-third or two-thirds of the search space. While this approach works, it does not improve on binary search for this problem since we need more API calls per iteration.

Algorithm

Initialize two pointers: l = 1 and r = n.
Calculate two midpoints: m1 = l + (r - l) / 3 and m2 = r - (r - l) / 3.
Check guess(m1) and guess(m2):
- If either returns 0, return that midpoint.
- If both results sum to 0 (one is 1, other is -1), the answer lies between m1 and m2; update both bounds.
- If guess(m1) is -1, the answer is below m1; set r = m1 - 1.
- Otherwise, the answer is above m2; set l = m2 + 1.
Repeat until the number is found.

# The guess API is already defined for you.
# @param num, your guess
# @return -1 if num is higher than the picked number
#          1 if num is lower than the picked number
#          otherwise return 0
# def guess(num: int) -> int:

class Solution:
    def guessNumber(self, n: int) -> int:
        l, r = 1, n
        while True:
            m1 = l + (r - l) // 3
            m2 = r - (r - l) // 3
            if guess(m1) == 0:
                return m1
            if guess(m2) == 0:
                return m2
            if guess(m1) + guess(m2) == 0:
                l = m1 + 1
                r = m2 - 1
            elif guess(m1) == -1:
                r = m1 - 1
            else:
                l = m2 + 1

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution extends GuessGame {
    public int guessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m1 = l + (r - l) / 3;
            int m2 = r - (r - l) / 3;
            if (guess(m1) == 0) return m1;
            if (guess(m2) == 0) return m2;
            if (guess(m1) + guess(m2) == 0) {
                l = m1 + 1;
                r = m2 - 1;
            } else if (guess(m1) == -1) {
                r = m1 - 1;
            } else {
                l = m2 + 1;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

class Solution {
public:
    int guessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m1 = l + (r - l) / 3;
            int m2 = r - (r - l) / 3;
            if (guess(m1) == 0) return m1;
            if (guess(m2) == 0) return m2;
            if (guess(m1) + guess(m2) == 0) {
                l = m1 + 1;
                r = m2 - 1;
            } else if (guess(m1) == -1) {
                r = m1 - 1;
            } else {
                l = m2 + 1;
            }
        }
    }
};

/**
 * Forward declaration of guess API.
 * @param {number} num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * function guess(num) {}
 */

class Solution {
    /**
     * @param {number} n
     * @return {number}
     */
    guessNumber(n) {
        let l = 1,
            r = n;
        while (true) {
            let m1 = l + Math.floor((r - l) / 3);
            let m2 = r - Math.floor((r - l) / 3);
            if (guess(m1) === 0) return m1;
            if (guess(m2) === 0) return m2;
            if (guess(m1) + guess(m2) === 0) {
                l = m1 + 1;
                r = m2 - 1;
            } else if (guess(m1) === -1) {
                r = m1 - 1;
            } else {
                l = m2 + 1;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * int guess(int num);
 */

public class Solution : GuessGame {
    public int GuessNumber(int n) {
        int l = 1, r = n;
        while (true) {
            int m1 = l + (r - l) / 3;
            int m2 = r - (r - l) / 3;

            if (guess(m1) == 0) return m1;
            if (guess(m2) == 0) return m2;

            if (guess(m1) + guess(m2) == 0) {
                l = m1 + 1;
                r = m2 - 1;
            }
            else if (guess(m1) == -1) {
                r = m1 - 1;
            }
            else {
                l = m2 + 1;
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(num int) int;
 */

func guessNumber(n int) int {
    l, r := 1, n
    for {
        m1 := l + (r-l)/3
        m2 := r - (r-l)/3
        if guess(m1) == 0 {
            return m1
        }
        if guess(m2) == 0 {
            return m2
        }
        if guess(m1)+guess(m2) == 0 {
            l = m1 + 1
            r = m2 - 1
        } else if guess(m1) == -1 {
            r = m1 - 1
        } else {
            l = m2 + 1
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * fun guess(num: Int): Int {}
 */

class Solution : GuessGame() {
    override fun guessNumber(n: Int): Int {
        var l = 1
        var r = n
        while (true) {
            val m1 = l + (r - l) / 3
            val m2 = r - (r - l) / 3
            if (guess(m1) == 0) return m1
            if (guess(m2) == 0) return m2
            if (guess(m1) + guess(m2) == 0) {
                l = m1 + 1
                r = m2 - 1
            } else if (guess(m1) == -1) {
                r = m1 - 1
            } else {
                l = m2 + 1
            }
        }
    }
}

/**
 * Forward declaration of guess API.
 * @param  num   your guess
 * @return 	     -1 if num is higher than the picked number
 *			      1 if num is lower than the picked number
 *               otherwise return 0
 * func guess(_ num: Int) -> Int
 */

class Solution : GuessGame {
    func guessNumber(_ n: Int) -> Int {
        var l = 1
        var r = n
        while true {
            let m1 = l + (r - l) / 3
            let m2 = r - (r - l) / 3
            if guess(m1) == 0 { return m1 }
            if guess(m2) == 0 { return m2 }
            if guess(m1) + guess(m2) == 0 {
                l = m1 + 1
                r = m2 - 1
            } else if guess(m1) == -1 {
                r = m1 - 1
            } else {
                l = m2 + 1
            }
        }
    }
}

// Forward declaration of guess API.
// unsafe fn guess(num: i32) -> i32;

impl Solution {
    unsafe fn guessNumber(n: i32) -> i32 {
        let (mut l, mut r) = (1, n);
        loop {
            let m1 = l + (r - l) / 3;
            let m2 = r - (r - l) / 3;
            if guess(m1) == 0 { return m1; }
            if guess(m2) == 0 { return m2; }
            if guess(m1) + guess(m2) == 0 {
                l = m1 + 1;
                r = m2 - 1;
            } else if guess(m1) == -1 {
                r = m1 - 1;
            } else {
                l = m2 + 1;
            }
        }
    }
}

Time & Space Complexity

Time complexity: $O(\log_3 n)$
Space complexity: $O(1)$

Common Pitfalls

Misinterpreting the guess() API Return Values

The guess() API returns -1 when your guess is higher than the picked number and 1 when your guess is lower. This is counterintuitive since many people expect positive to mean "go higher." Mixing up these return values causes the binary search to move in the wrong direction, leading to infinite loops or incorrect results.

Integer Overflow When Computing the Midpoint

Calculating the midpoint as (l + r) / 2 can cause integer overflow when l and r are both large values close to the maximum integer. The safe approach is to use l + (r - l) / 2, which avoids adding two large numbers together and prevents overflow in languages with fixed-size integers like Java, C++, and Go.