text: Logic
  
  # Operator Search: Level 3 Challenges
  
  $1 \ \boxed{\phantom{0}} \ 2 \ \boxed{\phantom{0}} \ 3 \ \boxed{\phantom{0}} \ 4 \ \boxed{\phantom{0}} \ 5$
  
  Use the mathematical operators $+,-,÷,×$ exactly once to fill in the blank boxes. What is the maximum real value that can be obtained?
  
  Using only the 4 basic arithmetic operations ($+, -, \times, \div$), what is the largest possible value of
  
  $\large 1\, \square\, 2\,\square \,3\, \square \, 4 ?$
  
  $\large 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8\ \Box \ 8 \ \Box\ 8=1000$
  
  What is the minimum number of operators that can be filled in the boxes to make the equation above true?
  
  Note: Boxes can be left blank to denote concetanation of adjacent digits
  
  \large{\begin{aligned} 1 &=& 44 \div 44 \\ 2 &=& 4 \times 4 \div (4 + 4 ) \\ 3 &=& (4 + 4 + 4) \div 4 \\ 4 &=& 4 + (4\times(4-4)) \\ 5 &=& (4 + (4\times4)) \div 4 \end{aligned}}
  
  Above shows the first 5 positive integers formed by using the four mathematical operators ($+ \ - \ \times \ \div$) only on the digit 4 four times.
  
  What is the smallest positive integer that cannot be represented using these conditions?
  
  Note: You are allowed to join the digits together: $44 + 44$.
  
  $\large 1 \; \underbrace{\square \; 2 \; \square \; \cdots \; \square \; n \; \square}_{n \text{ number of }\square\text{'s}} \; (n+1) = n+ 2$
  
  What is the minimum value of the positive integer $n>1$ such that we can fill in all the boxes above by using at least one of the four mathematical operators ( $+, -, \times , \div$ ) and the equation holds true?
  
  Note: Order of operations (BODMAS) applied.
  
  ×
explanation: The document contains a series of math problems, each with a specific question and no provided answers. The problems involve using arithmetic operations to achieve certain values or conditions.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Logic
  
  # Operator Search: Level 3 Challenges
  
  $1 \ \boxed{\phantom{0}} \ 2 \ \boxed{\phantom{0}} \ 3 \ \boxed{\phantom{0}} \ 4 \ \boxed{\phantom{0}} \ 5$
  
  Use the mathematical operators $+,-,÷,×$ exactly once to fill in the blank boxes. What is the maximum real value that can be obtained?
  
  Using only the 4 basic arithmetic operations ($+, -, \times, \div$), what is the largest possible value of
  
  $\large 1\, \square\, 2\,\square \,3\, \square \, 4 ?$
  
  $\large 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8 \ \Box \ 8\ \Box \ 8 \ \Box\ 8=1000$
  
  What is the minimum number of operators that can be filled in the boxes to make the equation above true?
  
  Note: Boxes can be left blank to denote concetanation of adjacent digits
  
  \large{\begin{aligned} 1 &=& 44 \div 44 \\ 2 &=& 4 \times 4 \div (4 + 4 ) \\ 3 &=& (4 + 4 + 4) \div 4 \\ 4 &=& 4 + (4\times(4-4)) \\ 5 &=& (4 + (4\times4)) \div 4 \end{aligned}}
  
  Above shows the first 5 positive integers formed by using the four mathematical operators ($+ \ - \ \times \ \div$) only on the digit 4 four times.
  
  What is the smallest positive integer that cannot be represented using these conditions?
  
  Note: You are allowed to join the digits together: $44 + 44$.
  
  $\large 1 \; \underbrace{\square \; 2 \; \square \; \cdots \; \square \; n \; \square}_{n \text{ number of }\square\text{'s}} \; (n+1) = n+ 2$
  
  What is the minimum value of the positive integer $n>1$ such that we can fill in all the boxes above by using at least one of the four mathematical operators ( $+, -, \times , \div$ ) and the equation holds true?
  
  Note: Order of operations (BODMAS) applied.
  
  ×
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of math problems, each with a specific question and no provided answers. The problems involve using arithmetic operations to achieve certain values or conditions.
  
  Label: multiple-question