Cube problems

[sarab]

i have searched the thread . got many answers but still the funda is not working for me.I want the approcah how to go abt sloving these problems.visualiztion is okie..but need some fromulas.If anyone can share how to go abt sloving these. i have the naswers but i get different answers if iapply the rules given in th some post .

questions are here follows :
1) 125 small but identical cubes have been put together to form a large cube.how many such small cubes would be required to cover this large cube compeltely?

2)what is the maximum number of identical pieces of cube that can be cut into by 3 cuts?


3)125 small but identical cubes are cut to form a large cube.This large cube is now painted on all six faces.
How many of the smaller cubes have no faces painted at all / have exactly one face painted?


4) What is the least number of cuts required to cut a cube into 24 indentical pieces?


5) It was found that a cube can be cut into ceratin number of identical cuboids each measuring 1 cm * 2cm *5 cm.what is the side of the smallest such sube.How many such cuboids can be formwed from such a cube?


6) A cube of side 6 cm, has cut into 64 smaller but identical cubes.If it was estimared that it would take a 4 ltrs of paint to paint all the faces of the original cube, then how much paint is the original cube, then how much paint is required to paint all the faces of all the smaller cubes?


Plz dont give me answers i already have and wd post shortly i need the procedure?

[GauravShah]

Originally Posted by sarab

1) 125 small but identical cubes have been put together to form a large cube.how many such small cubes would be required to cover this large cube compeltely?

Edited : stoopid mistuk

The size of the cube made using 125 small cubes is 5 x 5 x 5. After covering this the size of the cube will become (n+2) x (n+2) x (n+2) i.e. 7 x 7 x 7.

So total numbers of cubes in 7 x 7 x 7 = 343.

Extra number of cubes required = 343 - 125 = 218

Originally Posted by sarab

2)what is the maximum number of identical pieces of cube that can be cut into by 3 cuts?

I think its clear that the 3 cuts are to be put in the three different axis.

Next putting 1 cut, splits an edge into 2 equal halves, 2 cuts will split into 3 equal parts and so on.

Here 1 cut is out in each axis ... So new cube dimensions are 2 x 2 x 2. Total number of smaller cubes = 8

[GauravShah]

Originally Posted by sarab

3)125 small but identical cubes are cut to form a large cube.This large cube is now painted on all six faces.
How many of the smaller cubes have no faces painted at all / have exactly one face painted?

Using Formula

No faces painted = (n-2) x (n-2) x (n-2)

So since cubes is of 5 x 5 x 5. Number of cubes with no face painted = 3 x 3 x 3 = 27

1 face painted = 6 x (n-2) x (n-2)

So its = 6 x 3 x 3 = 54

Originally Posted by sarab

4) What is the least number of cuts required to cut a cube into 24 indentical pieces?

First find factors of 24
24 = 2 x 2 x 2 x 3

Now write the number (24) in terms of 3 factors of least possible value.

So its 24 = 2 x 3 x 4.

Cuts required = (a-1) x (b-1) x (c-1) = 1 x 2 x 3 = 6

PS: Try to understand the formula while using it for the first time. Once understood, just use it without having to think everytime

[GauravShah]

Originally Posted by sarab

5) It was found that a cube can be cut into ceratin number of identical cuboids each measuring 1 cm * 2cm *5 cm.what is the side of the smallest such sube.How many such cuboids can be formwed from such a cube?

Side of the cube is the LCM of 1,2,5 ... i.e. 10 cm

For number of cuboids,

Its = (Volume of Cube) / (Volume of each cuboid)

= (10 x 10 x 10) / (1 x 2 x 5) = 100 cuboids

Originally Posted by sarab

6) A cube of side 6 cm, has cut into 64 smaller but identical cubes.If it was estimared that it would take a 4 ltrs of paint to paint all the faces of the original cube, then how much paint is required to paint all the faces of all the smaller cubes?

Surface area of the BIG cube = 6 x ( 6 x 6 ) = 216 sq. cm

So 4 ltrs req. for 216 sq. cms. ... so 1 ltr can paint 216/4 = 54 sq cms.

64 = 4 x 4 x 4. So side of smaller cube = 6/4 = 1.5 cms.

Total surface area of smaller cubes = 64 x [ 6 x (1.5 x 1.5) ] = 864 sq cms.

ltrs of paint required = 864/79 = 16 ltrs

Hoping all answers are correct . Ask queries if any.

HTH

Gaurav.

[Eccentric]

Originally Posted by sarab

1) 125 small but identical cubes have been put together to form a large cube.how many such small cubes would be required to cover this large cube compeltely?

I think Gaurav made a midstake here.
Size of larger cube = 125^1/3 = 5

The size of zube to completely engulf the cube of size_5 = 7

Think of it like what size of square is required to cover a squzre of size 5 from all sides.
6 will not do.. we need square of size 7.

no of small cubes = 7^3 = 343
more needed = 218.

[j0]

yeah he did make a mistake, and couldn't the question be interpreted to mean just cover all the surfaces of the cube and hence we'd need the TSA?
6x5^2= 150

[Eccentric]

Originally Posted by sarab

3)125 small but identical cubes are cut to form a large cube.This large cube is now painted on all six faces.
How many of the smaller cubes have no faces painted at all / have exactly one face painted?

3) Again cosdier a square of size 5. how many sqaure are not exposed from any whwre. The central 3. If you extapolate these to 3d the central 3sized cube will be completely immersed.
So ans for no side painted = 3^3=27

4) Considering a surface of 5 sized square, How many will be exposed from just one side. The central 3 sized square. i.e. 9. So onall surface = 9*6 = 54

[made_for_iims]

Originally Posted by GauravShah

Using Formula


So its 24 = 2 x 3 x 4.

Cuts required = (a-1) x (b-1) x (c-1) = 1 x 2 x 3 = 6

PS: Try to understand the formula while using it for the first time. Once understood, just use it without having to think everytime

Shouldn't it be 24 = 2*2*2*3

so total number of cuts needed 1+1+1+2 =5 cuts

[made_for_iims]

Originally Posted by sarab

5) It was found that a cube can be cut into ceratin number of identical cuboids each measuring 1 cm * 2cm *5 cm.what is the side of the smallest such sube.How many such cuboids can be formwed from such a cube?


6) A cube of side 6 cm, has cut into 64 smaller but identical cubes.If it was estimared that it would take a 4 ltrs of paint to paint all the faces of the original cube, then how much paint is the original cube, then how much paint is required to paint all the faces of all the smaller cubes?

5.
side of smallest cube possible.............try to visualise in 2D n then 3D....

should be the L.C.M of sides (1,2,5) = 10
so the side of such cube should be 10*10*10

so number of cuboids = 10*2*5 = 100


6. generalise solution
Let the side be a (which is 6 in this case)...
so T.Area=6a^2

now new dimensions = a/4 * a/4 * a/4

so T.area of one cube = 6 * (a/4) * (a/4)

Total Paint needed = 64 * 6 * (a/4) * (a/4)

so it should be 4 times the paint originally needed
or 16 litres

[FreakazoiD]

Originally Posted by GauravShah

Using Formula

No faces painted = (n-2) x (n-2) x (n-2)

So since cubes is of 5 x 5 x 5. Number of cubes with no face painted = 3 x 3 x 3 = 27

1 face painted = 6 x (n-2) x (n-2)

So its = 6 x 3 x 3 = 54


First find factors of 24
24 = 2 x 2 x 2 x 3

Now write the number (24) in terms of 3 factors of least possible value.

So its 24 = 2 x 3 x 4.

Cuts required = (a-1) x (b-1) x (c-1) = 1 x 2 x 3 = 6

PS: Try to understand the formula while using it for the first time. Once understood, just use it without having to think everytime

Can you explain this formula ?

The way I would solve this is : (dont have a formula, just intuitive...)

The max no. of pieces I can get from one cut is 2. For each additional cut, I can make pieces in multiples of 2 (if I cut elsewhere, the no. of pieces decreases). SO I can get 4 pices from 2 cuts, 8 from 3 and 16 from 4. 5 cuts will give me a maximum of 32 pieces. But we need just 24 here, so I would say the answer is 5.

cheers,

the freak