Question

Direction: A cube is painted and then divided cut into 336 smaller but identical pieces by making the minimum number of cuts possible. All cuts are parallel to some face.

How many smaller pieces have no face painted?

• A. 144
• B. 124
• C. 120
• D. 108

Answer 1

The correct option is C 120

Number of identical pieces $336=8\times 7\times 6.$ Hence we need 7 cut's in Z-direction, 6 cuts in Y- direction, 5-cuts in X-direction.



Means our cube cut into 8 parts along Z-directions say
$n_Z=8,$ Similarly $n_Y=7,n_X=6.$

For total number of identical pieces we can say

$n_X\times n_Y\times n_Z=336$

$\left\{\right(n_X-2)+2\}\times \left\{\right(n_Y-2)+2\}\times \left\{\right(n_Z-2)+2\}$

Now look at table below:

No. of pieces	Formula	Answers for $6\times 7\times 8$ cube
3-face painted (Corner pieces)	$2^3$	8
2-face painted piece	$4\left[\right(n_x-2)+(n_y-2)+(n_z-2\left)\right]$	$4\left[\right(8-2)+(7-2)+(6-2\left)\right]=60$
Only one face painted pieces	$2\left[\right(n_x-2\left)\right(n_y-2)+(n_y-2\left)\right(n_z-2)+(n_z-2\left)\right(n_x-2\left)\right]$	$2\left[\right(8-2\left)\right(7-2)+(7-2\left)\right(6-2)+(8-2)\times (6-2\left)\right]$
Pieces no face Painted	$(n_x-2)(n_y-2)(n_z-2)$	$(8-2)(7-2)(6-2)=120$

$(8-2)(7-2)(6-2)=120$