How to increase 3-Dimensional thinking?

In summary, the person is struggling with visualizing 3D problems in classical mechanics, chemistry, and math for their upcoming engineering entrance exams. They are specifically having trouble with a problem involving a cube that is painted on all sides and then broken into smaller cubes, trying to determine the probability of a small cube having 2 or more sides painted. They have put in effort but are unable to fully visualize the problem and are seeking advice on how to approach it. Some suggestions include reducing the problem to a lower dimension and creating a sketch to aid in visualization.
  • #1
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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
 
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  • #2
Princu said:
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.
 
  • #3
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?
 
  • #4
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..
 
  • #5
Princu said:
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?
 
  • #6
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?
 
  • #7
Princu said:
Maybe they will themselves form a cube or cuboid?
Yes, they would make a cube of how many blocks on each side?
Princu said:
Wait are there going to be 44 cubes with one side painted red?
Can you explain how you arrived at 44?
 
  • #8
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
 
  • #9
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.
 
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  • #10
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).
 
  • #11
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
 

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1. How can I improve my spatial reasoning skills?

Spatial reasoning is the ability to visualize and manipulate objects in three-dimensional space. To improve this skill, you can practice solving puzzles and games that involve spatial reasoning, such as jigsaw puzzles or Rubik's cubes. You can also try mentally rotating objects in your mind or drawing objects from different angles to enhance your visualization abilities.

2. Are there any specific exercises to increase 3D thinking?

Yes, there are several exercises that can help improve your 3D thinking. These include mentally rotating objects, drawing objects from different perspectives, and solving 3D puzzles. You can also try building structures with blocks or creating 3D models using materials like clay or paper.

3. Can practicing 3D thinking improve my problem-solving skills?

Yes, 3D thinking is closely linked to problem-solving abilities. By improving your spatial reasoning skills, you can better understand and manipulate complex spatial relationships, which can be useful in problem-solving tasks. It can also help you think outside the box and come up with creative solutions to problems.

4. How does 3D thinking benefit daily life?

Having strong 3D thinking skills can be beneficial in many aspects of daily life. It can help with tasks such as packing a suitcase efficiently, understanding and following maps, and visualizing how furniture or objects will fit in a space. It can also aid in tasks like cooking, where you may need to mentally rotate and manipulate objects while following a recipe.

5. Can 3D thinking be learned or is it a natural ability?

While some people may have a natural inclination towards 3D thinking, it is a skill that can be learned and improved with practice. By regularly engaging in activities that involve spatial reasoning, you can strengthen your 3D thinking abilities and become better at visualizing and manipulating objects in three-dimensional space.

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