MathDB

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 the 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

Problem Statement