AMC10 2013 10A, Problem 14

A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3$.  How many edges does the remaining solid have?

Solution: We can use Euler's polyhedron formula that says that $F+V=E+2$.  We know that there are originally $6$ faces on the cube, and each corner cube creates $3$ more.  $6+8(3) = 30$.  In addition, each cube creates $7$ new vertices while taking away the original $8$, yielding $8(7) = 56$ vertices.  Thus $E+2=56+30$, so $E=\fbox{84}$

Answer: 84