Painted Cube Problem-Journal #18

  Painted Cube Problem:      
                                       

                               


When a 3*3*3 cube made up o 1*1*1 is dipped into red paint ,investigate the number of smaller cubes:
 with 3 faces ,2 faces,1 face,0 face painted red as a fuction of edge length of larger cube.


Edge length                   Number of smaller cubes with 3 faces painted

        X                                                                           Y
        1                                                                            0
        2                                                                            8
        3                                                                            8
        4                                                                            8
        5                                                                            8   
                                                                                                       

Edge of length                                   Number of smaller cubes with 2 faces painted

       X                                                                              Y                                                     
        1                                                                              0
        2                                                                              0                                                       
        3                                                                             12
        4                                                                             24
        5                                                                             36
        6                                                                             48

Y=12(X-2),X>1

Edge of length                                       Number of  smaller cubes with 1 face painted

           X                                                                              Y
            1                                                                              0
            2                                                                              0
            3                                                                              6
            4                                                                             24
            5                                                                             54

Y=6(X-2)^2  ,X>1

Edge of length                                     Number of smaller cubes with 0 face painted

           X                                                                              Y
           1                                                                               0
           2                                                                               0
           3                                                                               1
           4                                                                               8
           5                                                                              27

                                                                        
Y=(X-2)^3, X>1