Question

a painted cube which is cut into an identical pieces of 729 smaller with minimum number of cuts now how many smaller pieces have no painted face​

Answer 1

Step-by-step explanation:

If a cube is divided into 729 identical cubelets,

It will have 9 X 9 cubes in a plane of the cube.

Cubes having exactly one side painted in one face=(n-2) X (n-2) =49

But, we have 6 faces.

Therefore, the total numbers of cubes having only one side painted

=6 X 49=294  Cubelets

I hope it helps

Answer 2

Given :  a painted cube which is cut into an identical pieces of 729 smaller with minimum number of cuts

To Find :  how many smaller pieces have no painted face​

Solution:

if , a , b and c are the cut in each direction then

729  = (a+1) *( b+1) * (c+1)

AM ≥ GM

( a + 1 + b + 1 + c + 1 )/ 3 ≥ ∛(a+1) *( b+1) * (c+1)

( a + 1 + b + 1 + c + 1 )/ 3≥ ∛729

=>( a + 1 + b + 1 + c + 1 )/3  ≥ 9

=> a + b + c ≥ 24

Minimum number of cuts = a + b + c = 24

Hence a = b = c = 8 is  the minimum number of cuts

24 cuts

No Painted Face = ( 9 - 2)³

= 7³

= 343

343 smaller pieces have no painted face​

Additional Info :

For a cube of side n*n*n painted on all sides which is uniformly cut into smaller cubes of dimension 1*1*1,

Number of cubes with 0 side painted= (n-2)³

Number of cubes with 1 sides painted =6(n - 2)²

Number of cubes with 2 sides painted= 12(n-2)

Number of cubes with 3 sides painted= 8 (always)

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