Problem Solving

Magoosh

A hundred identical cubic boxes are currently arranged in four cubes: a single cubic box, a 2 x 2 x 2 cube, a 3 x 3 x 3 cube, and a 4 x 4 x 4 cube. These four are not touching each other. All outward faces are painted and all inward faces are not painted. These four cubes are going to be dismantled and reassembled as a flat 10 x 10 square. The top and all the edges of this 10 x 10 square must be painted, but there is no requirement for paint on the bottom. How many individual faces will have to be painted to accommodate the requirements of this new design?

A. 0
B. 5
C. 9
D. 16
E. 27

OA C.

100 identical cubes arranged in
Single cubic boxes - 1*1*1 cube
- 2*2*2
- 3*3*3
- 4*4*4
The single cubic box (1*1*1 cube )has paint on all six side .
The (2*2*2) cube has 8 boxes and each of the has paint on three sides which forms 8 corner pieces.
In the (3*3*3) cube there are eight corner pieces, twelve edges pieces with paint on two sides, 6 face pieces with paint on one sides and one interior pieces without paint.
The (4*4*4) cube has eight corner pieces, twenty-four edge pieces, twenty-four faces and eight interior pieces.
For the 10*10 flat square we will need four corner pieces that has point on three side, thirty-to edge pieces with point on two sides, and sixty -four middle pieces with point on one side.
Cubic shape (1*1*1)
Total =1
corners=-
edge=-
face=-
interior=-

Cubic shape ( 2*2*2)
Total=-
corners=8
edge=-
face=-
interior=-

Cubic shape (3*3*3)
Total=-
corners=8
edge=12
face=6
interior=1

Cubic shape (4*4*4)
Total=-
corner=8
edge=24
face=24
interior=8

Total (Total) = 1
Total (corner) = 24
Total (edge) = 35
Total (face) = 30
Total (interior) =9
We could use any of the 24 corner boxes to fill the corners of the 10*10 flat square also fill the edge from some of the 35 edge boxes, the remaining ones as well as the 30 face boxes will be used to fill in the center. The only boxes that will need to be painted are the nine interior boxes, one side each.


$$Answer=\ option\ C$$