text: An Even Dozen
  
  The surface of a standard soccer ball is covered with 20 hexagons and 12 pentagons. Interestingly, while we might vary the number of hexagons, the number of pentagons must always be 12.
  
  That’s because the Euler characteristic of a sphere is 2, so VE + F = 2, where V is the number of vertices, or corners, E is the number of edges, and F is the number of faces. If P is the number of pentagons and H is the number of hexagons, then the total number of faces is F = P + H; the total number of vertices is V = (5P + 6H) / 3 (we divide by 3 because three faces meet at each vertex); and the total number of edges is E = (5P + 6H) / 2 (dividing by 2 because two faces meet at each edge). Putting those together gives
  
  $\displaystyle V-E+F={\frac {5P+6H}{3}}-{\frac {5P+6H}{2}}+P+H={\frac {P}{6}},$
  
  and since the Euler characteristic is 2, this means P must always be 12.
explanation: The text extract discusses the geometric properties of a standard soccer ball, specifically the number of hexagons and pentagons on its surface. It explains why the number of pentagons must always be 12 using the Euler characteristic of a sphere and provides a mathematical derivation to support this claim. The extract contains a detailed explanation but does not present a specific question or answer a specific question.
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: An Even Dozen
  
  The surface of a standard soccer ball is covered with 20 hexagons and 12 pentagons. Interestingly, while we might vary the number of hexagons, the number of pentagons must always be 12.
  
  That’s because the Euler characteristic of a sphere is 2, so VE + F = 2, where V is the number of vertices, or corners, E is the number of edges, and F is the number of faces. If P is the number of pentagons and H is the number of hexagons, then the total number of faces is F = P + H; the total number of vertices is V = (5P + 6H) / 3 (we divide by 3 because three faces meet at each vertex); and the total number of edges is E = (5P + 6H) / 2 (dividing by 2 because two faces meet at each edge). Putting those together gives
  
  $\displaystyle V-E+F={\frac {5P+6H}{3}}-{\frac {5P+6H}{2}}+P+H={\frac {P}{6}},$
  
  and since the Euler characteristic is 2, this means P must always be 12.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract discusses the geometric properties of a standard soccer ball, specifically the number of hexagons and pentagons on its surface. It explains why the number of pentagons must always be 12 using the Euler characteristic of a sphere and provides a mathematical derivation to support this claim. The extract contains a detailed explanation but does not present a specific question or answer a specific question.
  
  Label: no-question