Answer :
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.
