A cube of side 12 cm is painted red on all the faces and then cut into smaller cubes, each of side 3 cm. What is the total number of smaller cubes having none of their faces painted?

• If a cubic block of n*n*n individual cubes is taken, and the outer layer of cubes is removed (which is essentially what you are doing in this question), you will be left with a block of (n-2)*(n-2)*(n-2) cubes.
  
  There will always be 8 cubes with 3 faces painted (the corner cubes)
  
  A cube has 12 edges
  Each edge will have (n-2) cubes that are not corner cubes
  There will be 12(n-2) cubes that have 2 faces painted.
  
  A cube has 6 faces
  Each face will have (n-2)^2 cubes that have only one face painted.
  There will be 6(n-2)^2 cubes that have only one face painted
  
  Unpainted cubes: (n-2)^3
  One face painted: 6(n-2)^2
  2 faces painted: 12(n-2)
  3 faces painted: 8
  
  (n-2)^3 + 6(n-2)^2 + 12(n-2) + 8
  = n^3 - 6n^2 + 12n - 8 + 6n^2 - 24n + 24 + 12n - 24 + 8
  = n^3 cubes in total
  
  In this question n=4
  Unpainted cubes: 2^3 = 8
  One face painted: 6*2^2 = 24
  2 faces painted: 12*2 = 24
  3 faces painted: 8
• 13 years agoHelpfull: Yes(22) No(6)
• there is a formula : n = 12/3 , (n-2)^3, now (4-2)^3= 8 ans
• 9 years agoHelpfull: Yes(3) No(0)
• 8 .cube is divides into 64small cubes....n since all faces are painted then only the inner middle cubes are not painted.....slice it into 3 cuts.....4Horizontal parts are separated....nw 2nd slice contions inner 4 small cubes wwhich are not painted n also 3rd slice 4 inner middle cubes are nt painted ..therefore ans is 4+4=8
  
  
• 9 years agoHelpfull: Yes(1) No(0)
• (12/3)^3=64
• 9 years agoHelpfull: Yes(0) No(0)
• 64 cube is having 12cm side ,so when it is cut into 3cm different cube
  there are 64 cubes
• 9 years agoHelpfull: Yes(0) No(0)
• 4
  the inside 4 cubes are not painted
• 9 years agoHelpfull: Yes(0) No(1)
• 8
  The center 8 cubes
• 9 years agoHelpfull: Yes(0) No(0)
• 8 small cubes
  
• 9 years agoHelpfull: Yes(0) No(0)
• 8
  Coz only outer once will be painted not the correct ones....
• 7 years agoHelpfull: Yes(0) No(0)