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Unit Cube Paint Undertaking (Posted on 2016-09-28) Difficulty: 3 of 5
N unit cubes are assembled to form a larger cube (N is a positive integer.)

In the larger cube thus assembled, some of the faces are then painted.

The cube is now taken apart and it is found that 271 of the unit cubes have paint on them.

Find N.

No Solution Yet Submitted by K Sengupta    
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Solution Considering all the possibilities | Comment 3 of 6 |
The table below shows the number of cubelets that will have paint on them for different size cubes painted in any of the 9 possible ways described here:

types:
1 single face
2 two opposite faces
3 two adjacent faces
4 three faces meeting at a corner
5 three faces not meeting at a corner
6 four faces leaving two opposite faces bare
7 four faces leaving two adjacent faces bare
8 five faces
9 all faces

When 1000 of the small cubes is assembled into the large cube and 3 faces meeting at a corner are painted there are 271 cubelets with paint on them.

        type:
     n       1    2    3    4    5    6    7    8    9
     
      8      4    8    6    7    8    8    8    8    8
     27      9   18   15   19   21   24   23   25   26
     64     16   32   28   37   40   48   46   52   56
    125     25   50   45   61   65   80   77   89   98
    216     36   72   66   91   96  120  116  136  152
    343     49   98   91  127  133  168  163  193  218
    512     64  128  120  169  176  224  218  260  296
    729     81  162  153  217  225  288  281  337  386
   1000    100  200  190  271  280  360  352  424  488
   1331    121  242  231  331  341  440  431  521  602
   1728    144  288  276  397  408  528  518  628  728
   2197    169  338  325  469  481  624  613  745  866
   2744    196  392  378  547  560  728  716  872 1016
   3375    225  450  435  631  645  840  827 1009 1178
   4096    256  512  496  721  736  960  946 1156 1352
   4913    289  578  561  817  833 1088 1073 1313 1538
   5832    324  648  630  919  936 1224 1208 1480 1736
   6859    361  722  703 1027 1045 1368 1351 1657 1946
   8000    400  800  780 1141 1160 1520 1502 1844 2168
   9261    441  882  861 1261 1281 1680 1661 2041 2402
  10648    484  968  946 1387 1408 1848 1828 2248 2648
  12167    529 1058 1035 1519 1541 2024 2003 2465 2906
  13824    576 1152 1128 1657 1680 2208 2186 2692 3176
  15625    625 1250 1225 1801 1825 2400 2377 2929 3458
  
 As a check of the table, for 1000 (a 10x10x10 cube):
 
 One side painted: 100 small cubes
 Two opposite sides painted: 200 small cubes
 Two adjacent sides painted: 200 - 10 = 100 + 10*9 = 190
 Three sides around a vertex painted:
    Every small cube except those within a 9x9x9 based at opposite vertex
      = 1000 - 9*9*9 = 1000 - 729 = 271
      
 Three sides including two opposite sides:
    Every small cube except those in an 8x9x10 set
      = 1000 - 8*9*10 = 280
      
 Four sides excluding two opposite sides:
    Every small cube except 10*8*8 core
      = 1000 - 640 = 360
      
 Four sides leaving two adjacent sides bare:
    Every small cube except 8x9x9
      = 1000 - 8*9*9 = 352
      
 Five sides:
    Every small cube except 8x8x9:
      = 1000 - 8*8*9 = 424
      
 All sides:
    Every small cube except 8x8x8
      = 1000 - 8*8*8 = 488
  
DefDbl A-Z
Dim crlf$, ct(25, 9)


Private Sub Form_Load()
 Form1.Visible = True
 
' types:
' 1 single face
' 2 two opposite faces
' 3 two adjacent faces
' 4 three faces meeting at a corner
' 5 three faces not meeting at a corner
' 6 four faces leaving two opposite faces bare
' 7 four faces leaving two adjacent faces bare
' 8 five faces
' 9 all six faces
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For s = 2 To 25
   n = s * s * s
   Text1.Text = Text1.Text & mform(n, "#######") & "  "
   ct(s, 1) = s * s
   ct(s, 2) = 2 * ct(s, 1)
   ct(s, 3) = 2 * ct(s, 1) - s
   ct(s, 4) = 3 * s * s - 3 * s + 1
   ct(s, 5) = 3 * s * s - 2 * s
   ct(s, 6) = 4 * s * s - 4 * s
   ct(s, 7) = 4 * s * s - 5 * s + 2
   ct(s, 8) = n - (s - 2) * (s - 2) * (s - 2) - (s - 2) * (s - 2)
   ct(s, 9) = n - (s - 2) * (s - 2) * (s - 2)
   For i = 1 To 9
     Text1.Text = Text1.Text & mform(ct(s, i), "#####")
   Next
   Text1.Text = Text1.Text & crlf
 Next s

 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function


  Posted by Charlie on 2016-09-28 15:29:47
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