Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To determine an unknown integer [tex]\( x \)[/tex] within the range [tex]\( 1 \leq x \leq 15 \)[/tex] by asking up to 4 questions of the form "Is [tex]\( x=y \)[/tex] ?", we can use a binary search approach. This method optimizes the number of questions needed to find [tex]\( x \)[/tex] efficiently.
### Step-by-Step Solution:
1. Initialize the Range:
- Set the initial lower bound to 1: [tex]\( \text{lower\_bound} = 1 \)[/tex]
- Set the initial upper bound to 15: [tex]\( \text{upper\_bound} = 15 \)[/tex]
- Initialize the number of guesses made: [tex]\( \text{guesses} = 0 \)[/tex]
- Initialize a flag to check if [tex]\( x \)[/tex] is found: [tex]\( x\_found = \text{False} \)[/tex]
2. Begin the Search:
- As long as [tex]\( x \)[/tex] is not found and the number of guesses is less than 4, continue the process.
3. Calculate the Midpoint:
- In each step, calculate the midpoint of the current range: [tex]\( \text{guess} = \left\lfloor \frac{\text{lower\_bound} + \text{upper\_bound}}{2} \right\rfloor \)[/tex]
4. Ask the Question and Get the Response:
- Ask if [tex]\( x \)[/tex] is equal to the midpoint value (guess).
- Based on response:
- If the response is "Yes", [tex]\( x \)[/tex] is the midpoint value.
- If the response is "x > y", update the lower bound to exclude values less than or equal to guess.
- If the response is "x < y", update the upper bound to exclude values greater than or equal to guess.
5. Update Bounds Based on the Response:
- Adjust the bounds accordingly:
- If "Yes": Set [tex]\( x\_found \)[/tex] to True, meaning we've found [tex]\( x \)[/tex].
- If "x > y": Update [tex]\( \text{lower\_bound} = \text{guess} + 1 \)[/tex].
- If "x < y": Update [tex]\( \text{upper\_bound} = \text{guess} - 1 \)[/tex].
6. Increment the Number of Guesses:
- Each iteration represents one question asked, so increment the guess count: [tex]\( \text{guesses} += 1 \)[/tex].
### Example Walkthrough:
Given that [tex]\( x = 10 \)[/tex]:
1. First Guess:
- Initial bounds: lower=1, upper=15
- Guess: [tex]\( \left\lfloor \frac{1 + 15}{2} \right\rfloor = 8 \)[/tex]
- Question: "Is [tex]\( x = 8 \)[/tex]?"
- Response: "x > 8"
- Update bounds: lower=9, upper=15
2. Second Guess:
- New bounds: lower=9, upper=15
- Guess: [tex]\( \left\lfloor \frac{9 + 15}{2} \right\rfloor = 12 \)[/tex]
- Question: "Is [tex]\( x = 12 \)[/tex]?"
- Response: "x < 12"
- Update bounds: lower=9, upper=11
3. Third Guess:
- New bounds: lower=9, upper=11
- Guess: [tex]\( \left\lfloor \frac{9 + 11}{2} \right\rfloor = 10 \)[/tex]
- Question: "Is [tex]\( x = 10 \)[/tex]?"
- Response: "Yes"
- [tex]\( x \)[/tex] is found.
### Final Outcome:
- Number of guesses made: 3
- [tex]\( x \)[/tex] was successfully found.
- The bounds at the end of the process: lower=9, upper=11
Thus, the result is:
```
(3, True, 9, 11)
```
Here, [tex]\( 3 \)[/tex] is the number of guesses, [tex]\( \text{True} \)[/tex] indicates [tex]\( x \)[/tex] was found, and [tex]\( 9 \)[/tex] and [tex]\( 11 \)[/tex] indicate the final lower and upper bounds.
### Step-by-Step Solution:
1. Initialize the Range:
- Set the initial lower bound to 1: [tex]\( \text{lower\_bound} = 1 \)[/tex]
- Set the initial upper bound to 15: [tex]\( \text{upper\_bound} = 15 \)[/tex]
- Initialize the number of guesses made: [tex]\( \text{guesses} = 0 \)[/tex]
- Initialize a flag to check if [tex]\( x \)[/tex] is found: [tex]\( x\_found = \text{False} \)[/tex]
2. Begin the Search:
- As long as [tex]\( x \)[/tex] is not found and the number of guesses is less than 4, continue the process.
3. Calculate the Midpoint:
- In each step, calculate the midpoint of the current range: [tex]\( \text{guess} = \left\lfloor \frac{\text{lower\_bound} + \text{upper\_bound}}{2} \right\rfloor \)[/tex]
4. Ask the Question and Get the Response:
- Ask if [tex]\( x \)[/tex] is equal to the midpoint value (guess).
- Based on response:
- If the response is "Yes", [tex]\( x \)[/tex] is the midpoint value.
- If the response is "x > y", update the lower bound to exclude values less than or equal to guess.
- If the response is "x < y", update the upper bound to exclude values greater than or equal to guess.
5. Update Bounds Based on the Response:
- Adjust the bounds accordingly:
- If "Yes": Set [tex]\( x\_found \)[/tex] to True, meaning we've found [tex]\( x \)[/tex].
- If "x > y": Update [tex]\( \text{lower\_bound} = \text{guess} + 1 \)[/tex].
- If "x < y": Update [tex]\( \text{upper\_bound} = \text{guess} - 1 \)[/tex].
6. Increment the Number of Guesses:
- Each iteration represents one question asked, so increment the guess count: [tex]\( \text{guesses} += 1 \)[/tex].
### Example Walkthrough:
Given that [tex]\( x = 10 \)[/tex]:
1. First Guess:
- Initial bounds: lower=1, upper=15
- Guess: [tex]\( \left\lfloor \frac{1 + 15}{2} \right\rfloor = 8 \)[/tex]
- Question: "Is [tex]\( x = 8 \)[/tex]?"
- Response: "x > 8"
- Update bounds: lower=9, upper=15
2. Second Guess:
- New bounds: lower=9, upper=15
- Guess: [tex]\( \left\lfloor \frac{9 + 15}{2} \right\rfloor = 12 \)[/tex]
- Question: "Is [tex]\( x = 12 \)[/tex]?"
- Response: "x < 12"
- Update bounds: lower=9, upper=11
3. Third Guess:
- New bounds: lower=9, upper=11
- Guess: [tex]\( \left\lfloor \frac{9 + 11}{2} \right\rfloor = 10 \)[/tex]
- Question: "Is [tex]\( x = 10 \)[/tex]?"
- Response: "Yes"
- [tex]\( x \)[/tex] is found.
### Final Outcome:
- Number of guesses made: 3
- [tex]\( x \)[/tex] was successfully found.
- The bounds at the end of the process: lower=9, upper=11
Thus, the result is:
```
(3, True, 9, 11)
```
Here, [tex]\( 3 \)[/tex] is the number of guesses, [tex]\( \text{True} \)[/tex] indicates [tex]\( x \)[/tex] was found, and [tex]\( 9 \)[/tex] and [tex]\( 11 \)[/tex] indicate the final lower and upper bounds.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
