A cube of side 5cm is painted on all its side. If it is sliced into 1

Question

A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

cano · Accepted Answer

I think the quickest way is to imagine in your mind 1 side of the cube. Understand that the border cubes will have more than one side painted, so the ones with only one side are the ones in the middle, in this case 3x3=9. Then multiply that by 6 sides. If you will make a diagram, make only one side and then multiply by 6. Don't waste the little precious time that you have.
See the picture below to understand it better. The green squares have adjacent surfaces painted. Only the blue ones have 1 side only painted.

Bunuel · Answer

Look at the image below: https://gmatclub.com/forum/download/file.php?id=23575 Little cubes with exactly one painted side will be those 3*3=9, which are in the center of each face. (6 faces)*(9 per each) = 54. Answer: D. Similar questions to practice: the-entire-exterior-of-a-large-wooden-cube-is-painted-red-155955.html a-big-cube-is-formed-by-rearranging-the-160-coloured-and-99424.html 64-small-identical-cubes-are-used-to-form-a-large-cube-151009.html a-wooden-cube-whose-edge-length-is-10-inches-is-composed-of-162570.html if-a-4-cm-cube-is-cut-into-1-cm-cubes-then-what-is-the-107843.html a-large-cube-consists-of-125-identical-small-cubes-how-110256.html 3-D Geometry Questions to practice: 3-d-geometry-questions-171024.html

wudy · Answer

D. 6*9=54. A cube has 6 sides, each of which has 9 cubes with one sides painted.
32223
21112
21112
21112
32223

sumande · Answer

9 such cubes for each of the 6 surfaces = 9 * 6 = 54 cubes. (D)

Andrey2010 · Answer

Dear Wudy and Sumande

Please make the solution more clear.

First I don't understand how to get integer from 5 ^ 1/3 to get dimension of cube.

Also if you're talking about 9 cubes for every surface (3 x 3 x 3) and we're looknig for cubes with exactly one side painted that the solution is 6. One middle cube has only one painted side on every surface.
Or I didn't undertand the condition. Please explain.

Thank you in advance.

r019h · Answer

are you sure it's a 5 cubic centimeter cube or is it a 5 cubic metre cube?? there should be five 1 cubic centimeter cubes when a 5 cubic centimeter cube is cut up

wudy · Answer

Well. I though the cube is 5*5*5 centimeter, or the question does not make sense-- how a 5 cubic centimeter cube is divided into 1 cubic centimeter cube?

The number listed above is how many faces painted for each 1cm*1cm cube on a surface.
32223 
21112 
21112 
21112 
32223

sumande · Answer

My assumption was exactly the same. I didn't think that a cube of side 5^(1/3) cm being divided into 1 cubic centimeter cubes makes sense. So, I assumed that each of the sides of the cube was 5 cm (a 5 cm cube as opposed to 5 cubic centimeter).

defenestrate · Answer

There is another way to do this problem. We know that all of the squares on each side have 2 sides (or 3 sides) painted. we want to remove them.

Make a new square which represents all of these squares. 5-2(one inch from each edge)=3

3.3=9 squares on each side.

9.6 sides is 54 squares

restore · Answer

A cube of side 5cm is painted on all of its sides. If it is sliced into 1 cubic cm cubes, how many 1 cubic cm cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

D. This came straight from the GMAT club math review, but I'm a bit tripped up on the explanation

puneet478 · Answer

The answer is 'D'.

A 5 cm^3 cube divided in 1cm^3 cubes = 125 cubes in total. A cube has 6 faces. 
On each face, all the cubes surrounding the boundary will have more than 1 side painted, which makes total of 5+5+3+3 = 16 cubes.

So we are left with 25-16 = 9 cubes on each face (which has only 1 side painted) totaling to 9*6=54 cubes.

Spidy001 · Answer

all the boundary cubes will have more than one face painted. only the center ones on each side of the larger cube, will have paint on just one side.

if we observer one face, there are 9 such cubes which have only one face painted.

so this times 6 faces we will have 9*6 = 54 such small cubes .

Answer is D.

gladvijay · Answer

the smaller cubes which will share an edge will have more than 2 faces painted.. so only the center cubes will have to be considered.. so .. 3*3 on each face.. 6 faces = 54

154238 · Answer

Hi.. is there any formula to calculate total number of cubes after division.
Let say a 6cm cube is divided into 2cm cubes ?

PareshGmat · Answer

Yes there is 
it will be 6^3 / 2 ^3 = 27 cubes made

mau5 · Answer

For those who have a problem in visualising they can atleast come to a conlusion that such cubes will be present on all the six sides. Now, whatever be this number(9 in this case), the answer will be a multiple of 6. Thus, just choose an option which is a multiple of 6. 
However, GMAT can give 2 options where in more than 1 option is a multiple of 6. In this question, luckily we have only one such option.

D.

snehamd1309 · Answer

Hi Bunuel, When u show the image then i can quickly make out the answer and it gets very easy but this kind of 3-D accurate figure cant be drawn on paper and difficult to be thought in mind within 2 minutes. What should be done??

What is the best way to solve such questions?

PareshGmat · Answer

Not Bunuel, but lets try..

Refer diagram below; just imagine one side of the cube cube.png It will have 5 * 5 = 25 squares of 1 units The border (the corner squares) will always be 2 side painted, so remaining are the pink shaded squares There calculation per side = (5-2) * (5-2) = 3*3 = 9 (Two rows & two Columns are to be subtracted) Total 6 sides = 9*6 = 54 Answer = 54

keyvan37 · Answer

Shouldn't it ask " A cube of side 5cm is painted on all its   faces  "

A cube of side 5cm is painted on all its side. If it is sliced into 1

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A cube of side 5cm is painted on all its side. If it is sliced into 1

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