Mark Haynes

 Profile Answers by Mark Haynes Questions by Mark Haynes

• Jan 12th, 2012

 

245 pieces have no color on them
98 pieces have black on one side

suneel

• Jan 23rd, 2012

 

125

Mark Haynes

 Profile Answers by Mark Haynes Questions by Mark Haynes

• Jan 30th, 2012

 

The question is;

"A solid cube is painted black on two of its opposite faces and is cut into 343 pieces. how many small pieces have no paint??"

Thus we know, we have;

A solid cube
- painted black on 2 of its opposite sides
- cut into 343 pieces

And we want to know;

How many small pieces have no paint?

Since its a cube;
we know that how tall it is equals how wide it is which equals how deep it is.

So in terms of 1 small piece, we know that the solid cubes dimensions must be:
7 small cubes tall by
7 small cubes wide by
7 small cubes deep,

since this equals 343 small pieces.

So, since we are only asking about pieces with paint on them, we know that the pieces inside the solid cube will have no paint on them, and that there 125 pieces that make up the inside of the cube

Now Suneel answer is partly correct there are AT LEAST 125 small pieces with no paint, however those 125 small pieces are not the only ones without paint on them.

Now for the tricky part:

the "cube is painted black on two of its opposite faces"

By this do we mean:
1. One face of the cube is painted black, and its opposite face is painted black
- hence, two of its opposite faces (one face on one side, the second face on the opposite side)
- thus, 49 small pieces on one face painted black
- and 49 small on the opposite face paited black
- for a total of 98 small pieces painted black.

If we start with the total of 343 small pieces,
and subtract the 125 small pieces we know have no black paint
we get 218 small pieces, and from these 218 small pieces we
subtract the 98 small pieces we know are painted black we come
up with an additional 120 small pieces with no black paint, and
that 120 small pieces added to the other 125 small pieces we know
are not painted black gives a total of 245 small pieces not painted
black.

Thus, 343 small pieces make up the cube:
245 small pieces are not painted black
98 small pieces have black on one side

- or do we mean -

2. One face of the cube is painted black and its opposite face is painted black,
then another face is painted black and its opposite face is painted black.
- hence, two of its opposite faces (one pair of opposite faces, and a second pair of opposite faces)
- thus, 49 small pieces on one face painted black
- and 49 small on the opposite face paited black,
- then another side is painted black,
- this adds 35 more small pieces painted black on one side
- and its opposite side is painted black,
- this adds another 35 more small pieces painted black on one side
- for a total of 168 small pieces painted black on at least one side

If we start with the total of 343 small pieces,
and subtract the 125 small pieces we know have no black paint
we get 218 small pieces, and from these 218 small pieces we
subtract the 168 small pieces we know are painted black on at
least one side we come up with an additional 50 small pieces with
no black paint, and that 50 small pieces added to the other
125 small pieces we know are not painted black gives a total
of 175 small pieces not painted black.

Nathiya

• Feb 21st, 2012

 

Answer for this question is 125
use this formula:(n-2) the whole cube.
here n=7(7 cube=343)
so (7-2) the whole cube=125.
use the following formula:
for exactly one face=6*(n-2) whole square
for exactly two face=12(n-2)

priyadharshini

• Feb 26th, 2012

 

125

Mark Haynes

 Profile Answers by Mark Haynes Questions by Mark Haynes

• Feb 27th, 2012

 

The question is;

The Given, part of the puzzle:

"A solid cube:-
1) is painted black
1a) on two of its opposite faces
2) and is cut into 343 pieces.

Question: how many small pieces have no paint??"

Now, one "face" - lets call it A - on the cube contains how many smaller cubes?

Lets check... count em!

001.002.003.004.005.006.007
008.009.010.011.012.013.014
015.016.017.018.019.020.021
022.023.024.025.026.027.028
029.030.031.032.033.034.035
036.037.038.039.040.041.042
043.044.045.046.047.048.049

One face has a total of 49 cubes!

Now, its opposite face - lets call it B - would not share an edge or corner with "face" A, so ...

Lets check how many small cubes it would have ...

001.002.003.004.005.006.007
008.009.010.011.012.013.014
015.016.017.018.019.020.021
022.023.024.025.026.027.028
029.030.031.032.033.034.035
036.037.038.039.040.041.042
043.044.045.046.047.048.049

The opposite face has a total of 49 cubes!

... Now put black paint on both "face" A and "face" B

... Sum them ... 49 + 49 = 98 small cubes with black paint.

... Total Number of small cubes = 343

... So, 343 - 98 = 245 small cubes with no paint on them.

125 is the wrong answer to the question, the formula(n-2) where n = 7 gives you only the total number of small cubes that form the inside of the 7x7x7 cube.

Mark Haynes

 Profile Answers by Mark Haynes Questions by Mark Haynes

• Feb 28th, 2012

 

Ok, lets try this one more time to explain why 125 cubes is the wrong answer.

We are told that the larger, over all, cube is made up of 343 smaller cubes.

With a little knowledge of math its possible to realize that you get 343 smaller cubes from:

7 cubed, meaning 7 to the power of 3, meaning 7 x 7 x 7

So, what is a Cube?

Cube: A a region of space formed by six identical square faces joined along their edges. Three edges join at each corner to form a vertex.

Cube Face: Also called facets or sides. A cube has six faces which are all squares, so each face has four equal sides and all four interior angles are right angles

Cube Edge: A line segment formed where two edges meet. A cube has 12 edges. Because all faces are squares and congruent to each other, all 12 edges are the same length.

Cube Vertex: A point formed where three edges meet. A cube has 8 vertices

Now, given that this cube in the question is made up of smaller cubes, then:

The over all large cube can be thought of as being made up of 4 layers, with the series of cubes nested inside the next.

1st layer: a 1 x 1 x 1 cube, inside the
2nd layer: a 3 x 3 x 3 cube, inside the
3rd layer: a 5 x 5 x 5 cube, inside the
4th layer: a 7 x 7 x 7 cube

and the Sum of all the smaller cubes from 1st, 2nd, 3rd, and 4th is... 343

Now lets decompose the 4th layer, to find out how many smaller cubes make up this layer.

Well, because its a cube it has 8 Vertex, so remove those....

Now with the Vertex removed, lets remove the edges...
we know that each edge must contain only 5 smaller cubes
and that there are 12 edges.

For a total of 60 small cubes forming the edges

Now, the remaining small cubes on the faces, each face has a square made up of 5 x 5 smaller cubes
Now, 5 x 5 is 25 small cubes per face, and there are 6 face for a total of 150 smaller cubes

So, we have:

008 : small cubes that form the Cube Vertex of the larger cube
060+: small cubes that form the Cube Edges of the larger cube
150+: small cubes that remain on the Cube Faces of the larger cube
---
218 : Total number of small cubes that form the 4th layer of the larger cube

So, how many small pieces have paint??

Paint, after all, can ONLY be on the 4th layer of small cubes that form the larger cube

Now, one Cube Face of the Larger Cube - lets call it A - contains how many smaller cubes?

Lets check... count em!

001.002.003.004.005.006.007
008.009.010.011.012.013.014
015.016.017.018.019.020.021
022.023.024.025.026.027.028
029.030.031.032.033.034.035
036.037.038.039.040.041.042
043.044.045.046.047.048.049

Cube Face "A" has a total of 49 smaller cubes!

Now, its opposite Cube Face - lets call it "B" - would not share a Cube Edge or Cube Vertex with Cube Face "A"

Lets check how many small cubes it would have ...

001.002.003.004.005.006.007
008.009.010.011.012.013.014
015.016.017.018.019.020.021
022.023.024.025.026.027.028
029.030.031.032.033.034.035
036.037.038.039.040.041.042
043.044.045.046.047.048.049

Cube Face "B" - opposite face from Cube Face "A" - has a total of 49 smaller cubes!

049 : smaller cubes that form the Cube Face "A" of the larger cube
049+: smaller cubes that form the Cube Face "B" of the larger cube, and is opposite Cube Face "A"
---
098 : Total number of smaller cubes with paint on them

So, for that 4th layer of smaller cubes that form the Larger Cube we have:

218 : Total number of small cubes that form the 4th layer of the larger cube
098-: Minus Total number of smaller cubes WITH PAINT on them
----
120+: Total number of smaller cubes WITHOUT PAINT on them on the 4th layer of smaller cubes forming the Larger Cube

Now, the 1st layer, made of a 1 x 1 x 1 smaller cube, and the 2nd layer, made of 3 x 3 x 3 smaller cubes, and the 3rd layer made of 5 x 5 x 5 smaller cubes contain NO PAINT as well.

001 : 1st layer of smaller cube
026+: 2nd layer of smaller cubes
098+: 3rd layer of smaller cubes
---
125 : Total of smaller cubes from layers 1, 2, and 3 inside the larger cube WITHOUT PAINT.

So,...

125 : Total of smaller cubes from layers 1, 2, and 3 inside the Larger Cube WITHOUT PAINT.
120+: Total number of smaller cubes WITHOUT PAINT on them on the 4th layer the Larger Cube
----
245 : Total number of smaller cubes WITHOUT PAINT

Or...

343 : Total number of smaller cubes forming the Larger Cube
098-: Total number of smaller cubes WITH PAINT on them
----
245 : Total number of smaller cubes WITHOUT PAINT

Thus, anyway you look at it and add it up, 125 is the wrong answer for the total number of smaller cubes without paint. The correct answer is 245 smaller cubes without paint.

chandan sharma

• Mar 13th, 2013

 

125

jagadeesh

• May 6th, 2013

 

245

rajanikant karan

• May 20th, 2013

 

(n-2)^3=unpainted blocks
(7-2)=5^3=125

rajkumar

• Aug 2nd, 2013

 

311

girik

• Jan 3rd, 2015

 

painted black=49(top)+49(bottom)=98
no paint=343-98=245