Question 15
Let $EFGH,$ $EFDC,$ and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8,$ and let $A$ be the eighth vertex of the cube. Let $I,$ $J,$ and $K$ be points on $\overline{EF},$ $\overline{EH},$ and $\overline{EC},$ respectively, so that $EI = EJ = EK = 2.$ A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE},$ and containing the edges $\overline{IJ},$ $\overline{JK},$ and $\overline{KI}.$ The surface area of $S,$ including the walls of the tunnel, is $m + n\sqrt{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$