How to increase 3-Dimensional thinking?

[Princu]

Alright.I am preparing for engineering entrance exams next year.Most of the standard questions of Classical mechanics requires you to imagine the whole scenario in 3D.Unless you are able to vizualize it,You can't solve the problem.I am able to vizualize 70-80% of the problems but my imagination totally goes blank in some questions.I am kinda weak at 3D imagination.
Same thing goes for some problems in Chemistry(Difficulty in vizualizing shapes of molecules in 3D,Solid State) and maths(Vector and 3D geometry)..My exams are due next year in April.
Any advice or suggestions to develop this skill will be greatly appreciated.

An example of the problem is-(From Mathematics)
A cube is painted red on all of its sides.It is then broken into 125 small cubes of same size.A cube is picked and is found to have one of its side painted.What is the probability that it will have 2 or more sides painted?..PLease Help

 

[phinds]

An example of the problem is-(From Mathematics)
A cube is painted red on all of its sides.It is then broken into 125 small cubes of same size.A cube is picked and is found to have one of its side painted.What is the probability that it will have 2 or more sides painted?..PLease Help

You need to show some attempt to do this on your own and we'll be happy to help when we see where you are stuck.

 

[mhelgeson]

Are there any blocks with no sides painted? Where are they located? What shape to they make? How many blocks have no side painted?

Are there any blocks with exactly one side painted? Where are they located? What shape do they make? How many blocks have exactly one side painted?

 

[Princu]

I am not able to completely vizualize the problem..The small cubes which are cut from the corners of the big cube will have 3 sides painted(they will be 8 in number).Also the small cubes which lie on the edge will have 2 sides painted while the cubes taken from the middle part of the bigger cube will have no sides painted..That's all..Unalbe to count the exact number of cubes which have one or two side painttedBelieve me,I have put in some serious efforts to solve it,But just unable to completely vizualize it..

 

[mhelgeson]

the cubes taken from the middle part of the bigger cube will have no sides painted.

What three-dimensional shape do the cubes taken from the middle part make?

 

[Princu]

Maybe they will themselves form a cube or cuboid.I suck in vizualizing this question..Wait are there going to be 44 cubes with one side painted red?

 

[mhelgeson]

Maybe they will themselves form a cube or cuboid?

Yes, they would make a cube of how many blocks on each side?

Wait are there going to be 44 cubes with one side painted red?

Can you explain how you arrived at 44?

 

[Princu]

Sorry..That was a wild guess.(12*3+8).If we take a bigger cube of side 5 units ,then the cubes having no sides painted will form a cube of side 4 units

 

[voko]

In any 3D problem, look for a way to reduce it to a smaller dimension problem.

In this particular case, you need to consider the faces of the cube, which are 2D.

 

[Ibix]

Draw it if you cannot visualise it. An isometric sketch of a cube (one with a corner towards you) will give you an idea of how to approach it.

Also: Khaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaan!

(That isn't relevant - sorry).

 

[ehild]

Ibix is right: make a sketch. See attached one.

It is a 4x4x4 cube. All the outside ones are painted at last on one side.

There are 8 corner-cubes, painted on 3 sides.

All edges have two corner cubes, so there are 4-2 cubes which are painted on two sides for each edge. There are 12 edges: 24 cubes painted on two sides. How many are there for an NxNxN cube?
On each side of the big cube there are 4x4 small cubes. If you remove the edge-cubes and corner-cubes, you have a 2x2 square, 4 cubes painted on one side only. There are 6 sides of the big cube: 6x4=24 small cubes painted on one side. How many are there in case of an NxNxN cube?

The number of painted cubes is 8+24+24=56.

If you "peel off" all the outer layers of painted cubes, you get a cube with edges of 2 units, a smaller cube of size 2x2x2=8. How many do you get for an NxNxN cube?

The number of all small cubes, painted and not painted is 56+8=64 =4x4x4, as it should be.

ehild