Cube problems

[GauravShah]

Originally Posted by sarab

Gaurav
i didnt get this logic.

suppose i need to cut the cube by 10 cuts ....
i can split it 2 ,3,5 therefore 3*4*6= 72 identical cubes ..but answer is 44
for 3 cuts and 4 cuts it comes correct as 8 and 12.


try for 13 too ...i didnt follow as u rightly said

The logic is to divide the number into 3 equals parts if possible and put that many cuts on each axis. Also remember, 1 cut = 2 parts, 2 cuts = 3 parts and so on.

Now for 10 cuts, we try to split 10 into 3 equals parts around 10/3 = 3.33 .
So it will be 3,3,4 ... hope u got this,

So we have to put 3 cuts on 2 axis and 4 on the remaining axis to get maximum number of "cuboids" .

We can't have smaller cube with different number of cuts on different axis.

To total number of smaller cuboids obtained = (3+1) x (3+1) x (4+1) = 4 x 4 x 5 = 80.

So answer for 10 cuts shud be 80 according to me.

If total number of smaller "cubes" are required. we can only use 9 cuts, 3 on each axis to have (3+1)^3 = 64 smaller cubes. Please check the answer again.

So for 13, we will have parts close to 13/3 = 4.33 ... i.e. 4,4,5

Smaller cuboids obtained = 5 x 5 x 6 = 150.

HTH

Others who got this, please chip in if you can simplify it further

Gaurav.

[FreakazoiD]

Originally Posted by GauravShah

The number needs to be in terms of 3 factors, coz the factors represents the 3 axis. So 5 will be wrong.

freak, yeah by 3 cuts on 3 axis, we can have 8 smaller pieces, but with 4 cuts we cannot have 16 smaller pieces.

Actually there is a reverse kind of a problem too, like how many smaller pieces can be made using 4 cuts.

Here you divide the number into three parts of close values.

4 can only be divided as 1,1,2.

So the parts in each axis is 2 , 2 , 3. Total smaller pieces = 2 x 2 x 3 = 12.

I know that maybe its still not clear, still am quite sure about this and please post your doubts if any

HTH

Gaurav.

Ok. I got it. I was wrong. The answer is 6

There need to be 4 pieces in X, 3 pieces in Y and 2 in Z. Hence the number of cuts required would be 3+2+1 = 6

I got confused by your formula (a-1)*(b-1)*(c-1). I guess it should be (a-1)+(b-1)+(c-1)
so if you need 'n' pieces, factorize 'n' into 3 factors such that the sum of the 3 factors is minimum (the 3 factors represent the no. of identical pieces in each axis) and then apply the formula.

eg. if you need 40 pieces, 40 = 2*4*5 (which gives 1+3+4=8cuts) and also 40 = 8*5*1 (7+4+0=11cuts). 8 will be the answer.

cheers,

the freak

[GauravShah]

Originally Posted by sarab

Gaurav
how do u genralise the above formul
no faces u said subtract by 2 okie it works
but for 1 face painted ? u have taken 6 *n-2 *n2 ?????

how u derived the same and how about 2 faces 2 faces and so on ???
do explain the answer and formula how got that?

Here are the formulae:

Taking 5 x 5 x 5 cube as the sample cube.

For no faces painted:

Its the smaller cube inside the big one. given by (n-2) x (n-2) x (n-2)

For the sample cube, its 3 x 3 x 3 = 27

For 1 face painted:

There are (n-2) x (n-2) faces on each surface of the a cube which has just 1 face painted. So for 6 faces, its 6 x (n-2) x (n-2)

For the sample cube, its 6 x 3 x 3 = 54

For 2 faces painted:

All the edge pieces (not corners) should be counted here. Since there are 12 egdes. So its 12 x (n-2)

For the sample cube, its 12 x 3 = 36

For 3 faces painted:

Just the corner cubes. So its 8 for any cube.

So for the sample cube too its 8.

HTH

Gaurav.

[rohit_vij]

for cube havin small cubes with n on each side a bigger cube can be formed using n+2 small cubes on each side.
so extra cubes needed wld be (n+2)^3 - n ^3
here it wld be 7^3-125 = 343-125

2)the answer is 2^n where n is number of cuts

3) if there are n cubes on all sides and all faces are painted then
no faces painted =( n-2)^3
one face painted =(n-2)^2 *6
two face = 12 * (n -2)
three faces = 8

4) as 24 lies between 16 and 32 takin higher ans wld be 5

6) 64 *6*1^2/(6*6^2)*4

diclaimer all may not be right (silly mistakes always happen)

Originally Posted by 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 rohit_vij

2)the answer is 2^n where n is number of cuts

4) as 24 lies between 16 and 32 takin higher ans wld be 5

disclaimer all may not be right (silly mistakes always happen)

Can u explain how we can have 16 pieces with 4 cuts .

Seems a common doubt on this thread but i feel maximum pieces with 4 cuts is just 12, as explained before

Gaurav.

[made_for_iims]

Edited by made_for_iims

[rohit_vij]

dear gaurav havent read others explanation so no comments here goes my reasoning
lets say there is a cube with one cut we can divide it into 2
now place both the cut pieces one upon the other and then cut so i get four new pieces
then i do same so for 3 cuts 8 pieces
then i try again for four cuts it wld be sixteen
hope u get it.

disclaimer all i say is not gospel truth

Originally Posted by GauravShah

Can u explain how we can have 16 pieces with 4 cuts .

Seems a common doubt on this thread but i feel maximum pieces with 4 cuts is just 12, as explained before

Gaurav.

[GauravShah]

Originally Posted by rohit_vij

dear gaurav havent read others explanation so no comments here goes my reasoning
lets say there is a cube with one cut we can divide it into 2
now place both the cut pieces one upon the other and then cut so i get four new pieces
then i do same so for 3 cuts 8 pieces
then i try again for four cuts it wld be sixteen
hope u get it.

disclaimer all i say is not gospel truth

according to me, one is not allowed to move the pieces, the cuts are suppose to be placed on the 3 axis of the cube.

Am pretty confident about it coz had solved an exercise dealing with cubes.

But guess it can be best clarified by the poster of the question if he has the official answer.

Gaurav

[sarab]

Originally Posted by j0

sarab post the answer to this one.. we don't seem to be agreeing on the interpretation.

dude answer is 128 total no of cubes.

[sarab]

uestions with answers are as 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?
Ans :218

2)what is the maximum number of identical pieces of cube that can be cut into by 3 cuts?
Ans 8
if 4 cuts are there answer is 12


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?
ans 27 and 54


4) What is the least number of cuts required to cut a cube into 24 indentical pieces?
ans 6


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 formed from such a cube?
ans 10 cm 100 cuboid


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?
ans 16 ltrs



source of question from Time material