MathDB

Cube

ABCDEFGH

, labeled as shown below, has edge length

1

and is cut by a plane passing through vertex

D

and the midpoints

M

and

N

\overline{AB}

and

\overline{CG}

respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form

\frac{p}{q}

, where

p

and

q

are relatively prime positive integers. Find

p+q

. [asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

Problem Statement