text: # Combinatorial circle placement
  
  Suppose we have area bound by $0\leq x\leq 2$,$0\leq y\leq 2$
  
  We choose 17 points, there should be at least two with distance max $d\leq\frac{1}{\sqrt{2}}$
  
  Tried to count area of $A_{17circles}=(\frac{d}{2})^2\pi*17=6.67588438887831$
  
  Then to take square $A_{square}=(\frac{d}{2}+2+\frac{d}{2})^2=7.32842712474619$
  
  There is also each corner to be rounded $A_{removecorners}=d^2-(\frac{d}{2})^2\pi=0.107300918301276$
  
  Which results in $A_{square}-A_{removecorners}=7.22112620644491$
  
  Still $7.22112620644491>6.67588438887831$ so I cant dismiss argument that one is able to fit 17 points so, that $max_{distance}\geq d$
  
  How to dismiss this argument? How to prove that if we put 17 points into 2 by 2 box, some of them bound to be closer than $\frac{1}{\sqrt{2}}$
  
  Hint: Divide the $2\times 2$ square into $16$ equal little squares, by drawing three equally spaced horizontal lines, and three vertical lines. Then use the Pigeonhole Principle.
explanation: The text extract presents a combinatorial problem involving the placement of 17 points within a 2x2 square area. The author attempts to use area calculations to determine if it is possible for all points to be at least a certain distance apart. The extract includes a detailed calculation and a hint suggesting the use of the Pigeonhole Principle to solve the problem.
  
  The text contains a question and a hint to solve it.
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: # Combinatorial circle placement
  
  Suppose we have area bound by $0\leq x\leq 2$,$0\leq y\leq 2$
  
  We choose 17 points, there should be at least two with distance max $d\leq\frac{1}{\sqrt{2}}$
  
  Tried to count area of $A_{17circles}=(\frac{d}{2})^2\pi*17=6.67588438887831$
  
  Then to take square $A_{square}=(\frac{d}{2}+2+\frac{d}{2})^2=7.32842712474619$
  
  There is also each corner to be rounded $A_{removecorners}=d^2-(\frac{d}{2})^2\pi=0.107300918301276$
  
  Which results in $A_{square}-A_{removecorners}=7.22112620644491$
  
  Still $7.22112620644491>6.67588438887831$ so I cant dismiss argument that one is able to fit 17 points so, that $max_{distance}\geq d$
  
  How to dismiss this argument? How to prove that if we put 17 points into 2 by 2 box, some of them bound to be closer than $\frac{1}{\sqrt{2}}$
  
  Hint: Divide the $2\times 2$ square into $16$ equal little squares, by drawing three equally spaced horizontal lines, and three vertical lines. Then use the Pigeonhole Principle.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract presents a combinatorial problem involving the placement of 17 points within a 2x2 square area. The author attempts to use area calculations to determine if it is possible for all points to be at least a certain distance apart. The extract includes a detailed calculation and a hint suggesting the use of the Pigeonhole Principle to solve the problem.
  
  The text contains a question and a hint to solve it.
  
  Label: question-and-answer