Batch 3 - Class 199 - Cross Sections, Cutting and Painting Cubes

• (536Dudeney - 326) I have three square blocks of side 16, 15 and 12 respectively, and I wish to construct a square of side 25, with least number of cuts. The cuts need not be straight, and can be of any shape. What's the least number of cuts to do this?
• Answer: Three cuts, 6 pieces. 

• Drawing cubes in 2D to reflect their 3D character, and visualization

• Cube Nets - 11 in all

• How many cubes do you see in the picture below

• Answer: You will see 6 cubes if you view the white faces as top of cube. You will see 10 cubes if you see the white faces as bottom of the cubes

• Interpret the following cube

• This is an impossible cube. Cover the left side and see what the right side looks like. And vice-versa.

• Show Escher's construction, as another impossible construction

• Cut off a sphere (or use an apple). We have just made a cross-section - what is the cross section? Is it the cut apple? No, its the surface that is created. Note that a cross section of a 3D figure is a 2D surface. Creating a cross-section entails creating one straight cut - you are not allowed to tilt or curve the knife. 
• What kind of cross-sections can we create by cutting a sphere?
• All cross sections of a sphere are circles even if you try to cut it at an angle
• Lets explore cross-sections of a cube (let kids explore a cube, and think about what cross-sections they can generate)
• Can you get a square cross-section?
• Rectangle? (Non-square)
• Triangle?
• Trapezoid?
• Circle?
• A polygon with more than 4 vertices?
• More than 6 vertices?
• Draw the correlation of this with how you actually see a cube, since you can only see 2D.
• Lets explore cross-sections of a cylinder. Draw all types of cross sections that you can generate
• Circle, Oval, Rectangle
• Design a cork so that it fits a square, a circular and a triangular hole snugly.
• Answer: Take a cylindrical cork standing on a face, and cut a triangle as you look at it. The base provides the circular fit, the cylinder's cross section provides square, and the triangle is the third.

•  Three cubes with volumes 1, 8, and 27 are glued together to obtain a solid polyhedron with minimal surface area. What is that surface area?
• Answer: The combined surfaces of the three cubes is 6(9 + 4 + 1) = 6·14 = 84. We can reduce the surface area of the polyhedron by 2·4 = 8 when we put the two larger cubes together. Then we can reduce the surface area of the polygon by 4 more by gluing the unit cube so that it has a face in common with each of the other two cubes. So the minimal surface area of the polyhedron is 84 − 8 − 4 = 72 square units. 

• A 5 by 5 by 5 cube is formed using identical 1 by 1 by 1 cubes. A number of the smaller cubes are removed by punching out the fifteen designated columns from front to back, top to bottom, and side to side. What percentage of the volume of the original cube remains following the removal of the fifteen columns?

• Answer 54.4% (Only 68 of surface cubes remain - inside is hollowed)