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# The large solid wooden cube above is composed of 8 smaller cubes of eq

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Math Expert
Joined: 02 Sep 2009
Posts: 51185
The large solid wooden cube above is composed of 8 smaller cubes of eq  [#permalink]

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06 Dec 2017, 21:57
00:00

Difficulty:

35% (medium)

Question Stats:

82% (01:48) correct 18% (01:33) wrong based on 27 sessions

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The large solid wooden cube above is composed of 8 smaller cubes of equal size, and the total surface of the large cube is painted purple. If each of the smaller cubes is then cut into 8 cubes of equal size, how many of the smallest cubes will have exactly 3 purple faces?

(A) 4
(B) 8
(C) 16
(D) 24
(E) 32

Attachment:

2017-12-07_0955.png [ 3.72 KiB | Viewed 629 times ]

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Re: The large solid wooden cube above is composed of 8 smaller cubes of eq  [#permalink]

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07 Dec 2017, 11:36

When we cut the large solid wooden cube into 8 smaller cubes of equal size,
all the eight cubes will have 3 of the 6 faces which are painted purple.

Hence, the total number of cubes with 3 purple faces will be 8(Option B)
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The large solid wooden cube above is composed of 8 smaller cubes of eq  [#permalink]

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07 Dec 2017, 14:51
Bunuel wrote:

The large solid wooden cube above is composed of 8 smaller cubes of equal size, and the total surface of the large cube is painted purple. If each of the smaller cubes is then cut into 8 cubes of equal size, how many of the smallest cubes will have exactly 3 purple faces?

(A) 4
(B) 8
(C) 16
(D) 24
(E) 32

Attachment:
2017-12-07_0955.png

This problem initially made me imagine a parade of horribles. Then I noticed: all 8 cubes now are corners.

Their being corners allows their three purple faces to show (such that the whole surface of the big figure is painted).

All cuboids have 8 corners, whether the cuboid is divided into 8 cubes or 64 cubes. Cut up 8 cubes that are also corners -- and you still have 8 corners. But you have preserved only one tiny cube per division that, because it is cut off the corner, is itself a corner just like the original.

That is, when each cube here is divided, only one tiny cube will be a corresponding corner, like the original, with three painted faces.

Example: The upper left front cube now, when divided by 8, will yield only one corner exactly like itself: the upper left front cube of ITS eight divided cubes.

The other bigger cubes will follow suit.

So the 8 larger cubes, when divided, end up having only one tiny cube -- a corresponding corner -- with exactly 3 purple faces.

(8 * 1) = 8

The large solid wooden cube above is composed of 8 smaller cubes of eq &nbs [#permalink] 07 Dec 2017, 14:51
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