Question

three different faces of a cube are painted in three different colours red green and black this cube is cut into 216 is smaller but identical cubes what is the least number of smaller cubes have exactly one face painted red and no other face painted ​

Answer 1

Answer:

5 and 3 are on opposite faces and 2&4 are on opposite faces. So in this case paint can be done on 1,4,6 or 1,5,6 or 2,3,4 or 3,4,5 and so on) . After cutting in 216 small cubes, each face will have have 36 cubes. Therefore number of cubes painted exactly one face will be 84.

Step-by-step explanation:

Answer 2

Answer:

The least number of smaller cubes have exactly one face painted red and no other face painted ​ is $92$

Step-by-step explanation:

Step 1:

Let us consider a cube having each side of 6 cm then its volume will be 216 cm^3

If we cut 226 cubes from it then each cube will be 1 x 1 x 1

Step 2:

16 x 6=96 will have one side colored

8 cubes have three sides colored

$4 * 4 * 4 =64$ are not colored  

Step 3:

4 on each edge have two sides colored so on 12 edges total with

Two sides colored$4 * 22=48$

Step 4:

So $total=96+8+64+48=216$

92 and 96.

$216= 6*6*6$

3 colored faces of a big cube mean $6*6*3=108$ small cubes have at least one side painted.

Now for only one side painted:

Considering the 3 painted sides having one common corner of the big cube, i.e. 3 intersecting edges. The number of small cubes at the intersecting painted edges adds up to$6+5+5=16$ with more than one painted side. So the minimum no. of small cubes with exactly one painted side is $108-16=92.$Considering the 3 painted sides having no common corner, i.e. only 2 intersecting edges. Multi-face-painted cubes are$6+6=12$ in number. So maximum no. of one-face-painted small cubes $= 108-12=96.$