Guess Number Higher or Lower

This problem can be solved using the Binary Search algorithm. Binary search is a divide-and-conquer algorithm used for searching a sorted array. Here, even though we don’t have an array, the nature of the problem (guessing a number between 1 and n) suits the binary search algorithm.

Here is how we can implement it:

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class Solution:
    def guessNumber(self, n: int) -> int:
        left = 1
        right = n

        while left <= right:
            mid = left + (right - left) // 2
            result = guess(mid)

            if result == 0:
                return mid
            elif result == -1:
                right = mid - 1
            else:
                left = mid + 1

        return -1

In the code, we start by defining two pointers, left and right, which represent the start and end of the range where we are searching. Then we start a loop that continues until left is greater than right.

In each iteration, we calculate the middle point mid and make a guess using the guess function. If the result is 0, it means our guess is correct and we return mid. If the result is -1, it means the picked number is less than mid, so we update right to mid - 1. If the result is 1, it means the picked number is greater than mid, so we update left to mid + 1.

This algorithm is efficient as it splits the problem size in half at each step, resulting in a time complexity of O(log n).

Identifying Problem Isomorphism

“First Bad Version” is similar to “Guess Number Higher or Lower”. Both problems involve finding a specific target in a sorted range using a function that gives you a hint about where the target is located.

In “Guess Number Higher or Lower”, you are playing a guessing game where the system has picked a number from 1 to n, and you make a guess. Your guess is passed to a function guess(int num), which returns -1 if the picked number is smaller, 1 if the picked number is larger, or 0 if you’ve guessed correctly.

Similarly, in “First Bad Version”, you are given an API isBadVersion(version), which tells you if a version is bad. It returns false if the current version is good (i.e., it is before the first bad version), and true if the current version is bad or any version after that. Your goal is to find the first bad version with the minimum number of calls to the API.

Both problems are solved using a binary search algorithm due to the sorted nature of the search space and the hint provided by the functions to narrow down the search. These problems are simple binary search problems, with “First Bad Version” being slightly simpler due to the function returning a boolean value, which can immediately be used to limit the search space.

While the core idea is the same, the specifics of the problems and the details of the provided functions mean the implementations will be slightly different.

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def guessNumber(self, n: int) -> int:
	lowerBound, upperBound = 1, n
	myGuess = (lowerBound+upperBound) >> 1

	while (res := guess(myGuess)) != 0:
		if res == 1:
			lowerBound = myGuess+1
		else:
			upperBound = myGuess-1
		myGuess = (lowerBound+upperBound) >> 1

	return myGuess

Problem Classification

Based on the problem statement and the rules of the game, this problem can be classified under the following categories:

Binary Search: The game is set up such that every guess gives us a hint whether the target number is higher or lower. This divides our search space into half after each guess, making binary search an ideal strategy to solve this problem.
Interactive Problems: This problem involves interaction with a ‘black box’ function (guess()), which is typical of interactive problems. You don’t have complete information and you must interact with the function to gain more information.
Game Theory: This problem also falls under game theory because it involves strategizing the best way to guess a number picked by another player.

These categories overlap and are not mutually exclusive. Binary Search is the core algorithmic strategy, but the problem’s context makes it an interactive game theory problem.

Language Agnostic Coding Drills

Understanding Binary Search: Understand how binary search works, it’s a core concept behind this problem. Binary search operates by selecting the middle element of a sorted list of elements and if the selected (middle) element is not the target, the half in which the target cannot lie is eliminated and the search continues on the remaining half.
Understanding Interactive Problems: Understand what interactive problems are and how you can work with them. In interactive problems, you have to interact with an unknown function or environment to get additional information for the problem.
Handling the API Call: Learn how to interact with a given API or function. Here, we are interacting with a guess() function. We don’t know its internal workings but we know what it returns based on the input we provide.
Updating Lower and Upper Bounds: Depending on the result from the guess() function, you have to correctly update the bounds of your search space. This is an important step in binary search. If the guess is too high, the upper bound is reduced, and if the guess is too low, the lower bound is increased.
Converging to the Solution: Learn how to converge to a solution in an interactive problem. Here, the solution is reached when the guess() function returns 0, indicating that the guess is correct.

Problem-solving approach:

Start by setting the initial lower and upper bounds, which are 1 and n, respectively.
Make the first guess as the middle of these bounds.
Interact with the guess() function to find out whether the guessed number is too high or too low.
Based on the response, adjust the lower and upper bounds.
Make the next guess as the middle of the new bounds.
Continue this process until the guess() function confirms that the guessed number is correct. That’s when you’ve found the picked number.

Targeted Drills in Python

Understanding Binary Search:

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# Implement binary search in a sorted array
def binary_search(arr, target):
    low, high = 0, len(arr) - 1
    while low <= high:
        mid = (low + high) // 2
        if arr[mid] == target:
            return mid  # Found target
        elif arr[mid] < target:
            low = mid + 1  # Target is in the upper half
        else:
            high = mid - 1  # Target is in the lower half
    return -1  # Target not found

Understanding Interactive Problems:

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# Here, we will interact with a simple function that returns -1, 0, or 1 
# based on comparison with a pre-defined number
def guess(num):
    PICKED = 5
    if num == PICKED:
        return 0
    if num < PICKED:
        return 1
    if num > PICKED:
        return -1

# Interact with guess function
print(guess(3))  # Output: 1
print(guess(5))  # Output: 0
print(guess(6))  # Output: -1

Handling the API Call:

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# No separate drill for this as we have already implemented this in step 2.

Updating Lower and Upper Bounds:

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# We will use the guess function from step 2 here
def guess_number(n):
    lower, upper = 1, n
    while lower <= upper:
        mid = (lower + upper) // 2
        res = guess(mid)
        if res == 0:
            return mid
        elif res == 1:
            lower = mid + 1
        else:
            upper = mid - 1
    return -1  # Should never reach here in this problem

# Let's check with n = 10
print(guess_number(10))  # Output: 5 (As we set PICKED = 5 in guess function)

Converging to the Solution:

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# The final solution converges when we find the picked number.
# So, the final drill is same as the previous one. We already have the final solution in drill 4.