Guessing a Random Number Between 1 and 100

Solution & Explanation

The problem of guessing a random number between 1 and 100 can be effectively approached using a binary search strategy. This method is efficient because it systematically reduces the search space by half with each guess, ensuring that the number is found in a minimal number of attempts.

Basic Strategy: Binary Search

1. Initial Guess: Start by guessing the middle value of the range, which is 50. This divides the possible numbers into two equal halves.

   • If the guess is correct, you're done.
   • If not, you will be informed whether the target number is higher or lower.

2. Subsequent Guesses: Depending on whether the number is higher or lower than your guess:

   • Higher: Narrow the search to the upper half of the current range (e.g., 51 to 100 if starting from 50).
   • Lower: Narrow the search to the lower half of the current range (e.g., 1 to 49 if starting from 50).

3. Iterative Process: Continue this process of guessing the middle of the current range and adjusting the range based on feedback. This will systematically reduce the number of possibilities.

4. Conclusion: By the 7th guess, you will have narrowed the possibilities down to a single number.

Mathematical Justification

• The choice of 7 guesses is derived from the binary search tree depth needed to cover 100 numbers. Since 2^7 = 128, which is greater than 100, this ensures that the number can be found in at most 7 guesses.

Adapting to Challenging Numbers

If you have prior knowledge that the number is likely to be near the extremes (like 1 or 100), you can adjust your strategy:

1. Initial Adjustment: Instead of starting at the exact middle, consider starting slightly off-center towards the expected bias. For example, if numbers near 1 are more common, start at 25 instead of 50.

2. Subsequent Guesses: Use the same binary search approach but adjust the midpoints to account for the skew in distribution. This might mean taking more aggressive steps towards the extremes based on prior knowledge.

Example Scenario

• First Guess: 50
  • Feedback: "Higher"
• Second Guess: 75
  • Feedback: "Higher"
• Third Guess: 88
  • Feedback: "Higher"
• Fourth Guess: 94
  • Feedback: "Higher"
• Fifth Guess: 97
  • Feedback: "Higher"
• Sixth Guess: 99
  • Feedback: "Higher"
• Seventh Guess: 100
  • Feedback: "Correct"

This example illustrates how the binary search method efficiently narrows the possibilities, leveraging each piece of feedback to make informed subsequent guesses. By adhering to this structured approach, you can ensure that the number is guessed correctly in a minimal number of attempts, even when faced with challenging distributions.