Batch 3 - Class 188 - Lines & Squares

• Complete your crossword if you have not been able to complete it in class.
• (536Dudeney - 315) We have two cubes made of same material, one slightly (say 1%) larger than the other. We cut a hole through one so that the other cube can pass through it. However, the larger cube is still heavier than the smaller one. How is it possible?
• The hole has been made in the smaller cube and not the larger one! If you hold the smaller cube so that its diagonal is perpendicular to the ground, you see a hexagon, and a cube can be inscribed in it. This cube can be larger than the original cube (SQRT(6)-SQRT(2)=1.035). Later it has been shown that a slightly larger cube can be passed with a different orientation (upto about 1.06066 or 3SQRT(2)/4). This is also called Price Rupert's Cube.

• Consider the following grid. It has 10 straight lines and 17 squares.

• Here are 9 straight lines and 20 squares

• We want to find out smallest number of lines to make 100 squares

• Lets start from smaller numbers. What is the minimum number of lines required to create between 1 to 5 squares?
• How do you arrange these?
• What pattern do you notice?
• Can you extend this to find better answers to 17 and 20 squares relative to those given above?
• What kind of arrangement maximizes the number of squares for given number of lines?
• What if there are even number of lines or odd?
• What is the pattern in number of squares and number of lines required?
• Do all possibilities of number of squares get covered with this?
• Can you provide a lower and upper bound on number of lines required for 100 squares using this exercise?
• What if we didn't want to limit to square grids, and allowed rectangular grids
• What if you took the first grid with 10 lines, and added a line to it? Along the row? Along the column? How many squares do you add?
• What if you took a square grid and reduced lines from it? Assuming a line at the edge? 
• Can you generalize how many squares and lines a mxn regular rectangular grid may have? Alternately, list down the number of squares with few combinations of rows and columns (and hence lines).
• So now lets get back to 100 squares. What could be the approaches?
• Try 5x5, 6x6, 7x7 - what are the maximum number of squares possible
• Does that allow you to bound the result
• In above cases, what does an incremental line do? What does another incremental line do (in each direction)? Can you compose them to find the solution?
• There are other solutions possible. For example what happens if you cut a corner square from a grid - how many squares are eliminated? Can you start with a grid and eliminate squares? (Note that the starting grid would be a 6x7 grid in this case)
• What if you wanted 99 squares - can you use the "take off the corner method"?
• Some more advanced musings
• What if you were removing a line in the middle of the grid, starting with a square or rectangular grid?
• What if the grid was not being drawn on a plane, but rolled around a cylinder? How many squares result?
• What if you were allowed to tilt the line rolling around the cylinder slightly, so that the same line would give you many different "rows"?
• What if you drew this on a torus?
• Can you start to think about triangles instead of squares? How do you stack them up, and how many triangles do you get with certain number of lines?

• (536Dudeney - 319) The fly at point A below take a minute to crawl an edge of the cube. How long will it take it to reach point B?

• Answer: 2.236 minutes along the diagonal