Batch 2 - Class 27 - Solid Geometry and Packing Spheres

• Let kids demonstrate their planiverse machines

• MartinShColl 2.4 - Three points are selected at random on a sphere's surface. What is the probability that they fall within the same hemisphere?

  Answer: 1 (It is a certainty)
• MartinColl Page 135: There are four unit circles centered at vertices of a unit square. What is the area of the blank space in between, in terms of area of square and circle?

  What would be the similar area for space inside three unit circles centered at vertices of unit equilateral triangle?

• FirstMensa - 3, Martin Coll - Page 174: Picture Below - which of the papers in enclosed image can be folded into a cube (Answer: A,B,D)

• The figure on right shows all the 11 possible Hexominoes

• Triangular packing and triangular numbers
• Square packing and square numbers

• Relationship between triangular and square numbers

• FirstMensa - 94+ - How many cannon balls are there in a pyramid with front face showing 4,3,2,1 stack of balls

  Answer: 30How many in a 3 sided pyramid (Answer: 20)
• Hexagonal Close packing
• Cubic Packing (74% packing efficiency - still not proven to be optimal)

  Tetrahedral pyramidsSquare pyramids (layers run parallel to sides)

• At many train stations, post offices and courier services around the world, the cost of sending a rectangular box is determined by the sum of its dimensions; that is, length plus width plus height. Can you "cheat" by packing a box into a cheaper box? The assumption here is that the cost increases as length + width + height increases.

  Note that you can pack a box of greater lengthTry with 2D rectangle in rectangle
• MartinColl Page 173 - Take a cube. Draw a square hole in the cube. Can you now pass a cube larger than original cube through this square hole?

  Answer: Yes, 3/4*sqrt2 size for original unit cube