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The students in a school are having a cup stacking contest. John has a pyramid 9 layers high. How many cups will he need to add to his pyramid to increase

the number of layers to 12?


You need 18 cups little boy
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If the stacking is done as per that method, then we need, for a stack of L levels,

  total number of cups at level  L = L
  total number of cups for all levels from 1 to L :  1 + 2 + 3 + ... + L
             = L ( L + 1)/2
  So the number of cups needed to increase the number of layers to 12 from 9 is:

         12(12+1)/2 - 9 (9+1)/2 =  78 - 45 = 33

Another method is:
  We need L more cups to increase the number of layers from L - 1 to  L.  Hence, we need, to go from level 9 to level 12:  10+11+12 = 33 cups more.

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