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Question

Direction: A cube is painted and then divided cut into 336 smaller but identical pieces by making the minimum number of cuts possible. All cuts are parallel to some face.

How many smaller pieces have exactly 3 painted faces?

A
20
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B
8
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C
12
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D
6
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Solution

The correct option is B 8
Number of identical pieces 336=8×7×6. Hence we need 7 cut's in Z-direction, 6 cuts in Y- direction, 5-cuts in X-directions.



Means our cube cut into 8 parts along Z-directions say
nZ=8, Similarly nY=7,nX=6.

For total number of identical pieces we can say

nX×nY×nZ=336

{(nX2)+2}×{(nY2)+2}×{(nZ2)+2}

Now look at table below:
No.of pieces Formula Answrs for
6×7×8 cube
3-face painted
(Corner pieces)
23 8
2-face painted
piece
4[(nx2)+(ny2)+(nz2)] 4[(82)+(72)+(62)]=60
Only one face
painted pieces
2[(nx2)(ny2)+(ny2)(nz2)+(nz2)(nx2)] 2[(82)(72)+(72)(62)+(82)(62)]
Pieces no face
Painted
(nx2)(ny2)(nz2) (82)(72)(62)=120

All pieces from corners of cube = 8


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