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 |
Considering all the possibilities
