3cm x 3cm x 3cm cube is made of 27 small cubes, each measuring 1cm x 1cm x 1cm. Exterior is painted. How many of the small cubes have exactly one face painted?

Question

part b) if a 4 by 4 by 4 cube is assembled the same way. how many small cubes will have exactly one face painted? 

c) 5 x 5 x 5???

6 x 6 x 6???

Answer 1

Hi Julia

when exterior face of large cube is painted the small cubes at the edges will have at least 2 faces painted but the middle once will get one face painted we can calculate them 

 

1- large cube 3cm*3cm*3cm has small 3 cubes of 1cm ,  small cubes on the edge = 8 (one face of large cube)

    which is    1st edge=3,   +    2nd edge=2,  +   3rd edge=2,   +   4th edge=1    =8  (edge cubes of one face)

   number of cubes in one face is 9  (3*3) then 9-8 =1 is the cube that its one face painted, multiply by 6 faces will        get 6 small cubes have one face painted only 

2- large cube 4cm*4cm*4cm has small 4 cubes of 1cm ,   small cubes on the edge = 12 (one face of large cube)

    which is 1st edge=4,    +    2nd edge=3,   +   3rd edge=3,   +    4th edge= 2 =12(edge cubes of one face)

    number of cubes in one face is 16( 4*4)  when 16-12 =4 are the cubes which one face is painted, multiply by 6             faces will get 24 cubes have one face painted only 

3- with 5*5*5 will get 9  cubes one face painted, multiply by 6 will get 9*6= 54 cubes one face painted

4- with 6*6*6 will get 16  cubes one face painted, multiply by 6 will get 16*6=96 cubes one face painted

 

if u noticed there is strange relation between number of cubes in 1st edge and number of cubes in 4th edge which is: 

number of 1st edge cubes - number of 4th edge cubes always equal 2

such as in 3*3*3 cube 3-1 =2 and in 4*4*4 cube 4-2 =2, and in 5*5*5 cube 5-3=2 and in 6*6*6 is 6-4=2 also 

so we can use that to find number of cubes in 4th edge for any number of small cubes 

and the number of one face painted cubes (in one face of large cube) is square of that number (number of cubes in 4th edge) 

so for example 5*5*5 cube, its 4th edge has 5-2 =3 cubes then 3² =9 is number of one face painted cubes in one face of the large cube and 9*6 = 54 is the total number of the cubes that has one painted face.

also for 6*6*6 cube , its 4th edge has 6-2 = 4 cubes , then 4² =16 is number of one face painted cubes in one face of the large cube, and the total number of one face cubes is 16 *6 = 96  

for 10*10*10 cube the 4th edge is 10-2=8, then 8²=64 number of one face painted cubes in one face of large cube

64 *6 = 384 total number (of one face painted) cubes.

 

one face calculation

number cubes             edges cubes               one painted cubes                      

        3²              -              8                =             1²                                     

        4²              -              12              =             2²                                                    

        5²              -              16              =             3²

        6²              -              20              =             4²

 

to find total number of one face painted cubes multiply by 6 ( number of cube faces)

 

 

Best 

 

 

 

Answer 2

The 3 x 3 x 3  big cube, has six faces.  Only the small cube located in the center of each large cube face get one side (only) painted.  So the answer is 6.

 

The 4 x 4 x 4 big cube also has six faces.  However, the group of 4 small cubes in the center of a face now get one face (only) painted.  So the answer is 6 x 4 = 24.