Batch 3 - Class 225 - Slicing Planes

• (Dudeney 365) There are two jars of 10 liters full of water, and two empty ones of 5 and 4 liter capacity. You want to have 3 liters each in the smaller jars, without wasting any water. Devise the sequence of steps.
• Answer: 

• A plane is divided into regions by straight lines. Show that it is possible to color the regions with two colors so that adjacent regions are never the same color. Adjacent regions are defined as regions that share a side. 
• Instructor Notes: Let kids experiment with some regions and try to color. This may help them develop intuition
• What method of proof may be applicable here?
• Answer: Proof by induction. The base case of a region being divided into two by one line is trivial. Suppose a region with n lines satisfies this condition. Now draw the (n+1)th line. Invert the coloring on one side of this line. QED.

• A square is inscribed with a circle, which is inscribed with another square. Find the ratio of areas of smaller to larger square.

• Answer: 1/2. While the problem can be solved algebraically, the following visual proof suffices

• Explore with kids whether the number of regions always has to be the same, given a number of cuts
• For example, with 2 cuts, how many regions can we get? 3 or 4 - we lose a region if we have parallel lines
• With 3 cuts, how many can we get? 4, 6 or 7 - we lose a region if more than 2 cuts pass through the same point
• What do we do to maximize the number of regions
• Answer: Ensure every line cuts every other line, and no three line pass through the same point
• So not, lets try to solve smaller problems
• What the maximum number of slices we can get with 2 cuts - Answer: 4
• What the maximum number of slices we can get with 3 cuts - Answer: 7
• What the maximum number of slices we can get with 4 cuts - Answer: 11

• What is the pattern above?
• One way to look at the pattern is that amongst successive numbers, the successive difference goes up by 1. Can you explain this?
• Answer: If you already have 4 lines and you are drawing the fifth line, then this fifth line will intersect each other line once, and for each intersection, it will create a new region once
• Another way to look at it is that f(n+1) = 2.f(n) - nC2. Can you explain this?
• Answer: If we were able to divide each region, we would have 2.f(n). But for any pair of lines, a new line can only cut across three out of four regions formed by those two lines. So we lose one region for each pair of lines
• So what is f(10)? f(20)?
• Can you find a simple formula for f(n)?
• Answer: Sum of first n numbers + 1
• Can you find an intuitive logic for this simple formula?

• Imagine an infinite chessboard that contains a positive integer in each square. If the value in each square is equal to the average of its for neighbors to the north, south, west and east, what should all the values look like?
• Answer: They are all equal. Proof by contradiction: assume that for any square, its value is different from one of its neighbors. Then there must exist a neighbor which has lower value. Now think about that neighbor - since the original square was larger, this neighbor must not have a neighbor which is even smaller. Thus, this chain must lead to a scenario where the numbers start to turn negative. Since this is not possible, all numbers must be equal.