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Problem #18 2016 AMC 10: Labelling vertices of a cube

Source: AMC 10 2016 Problem #18 AMC 12 Problem # 14

February 3, 2016
AMC 10geometry3D geometrycombinatoricsAMCAMC 122016 AMC 10A

Problem Statement

Each vertex of a cube is to be labeled with an integer 11 through 88, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>3<spanclass=′latex−bold′>(C)</span>6<spanclass=′latex−bold′>(D)</span>12<spanclass=′latex−bold′>(E)</span>24<span class='latex-bold'>(A) </span> 1\qquad<span class='latex-bold'>(B) </span> 3\qquad<span class='latex-bold'>(C) </span>6 \qquad<span class='latex-bold'>(D) </span>12 \qquad<span class='latex-bold'>(E) </span>24
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