Batch 3 - Class 140 - More visual puzzles

• (536Dudeney 295) A train was going from Delhi to Mumbai. A wise mathematician claims that while most of the train is going from Delhi to Mumbai, parts of it are all the time going to Delhi! Seriously!? You mean when we cycle from home to school, parts of us are going home? Hey, we didn't say anything about cycling. Can you figure out what's going on?
• Answer: Because there is a part of the train's wheel that is actually below the line of motion (i.e. the rails), those part in fact are always going backwards. The path traced by these is called a Trochoid curve (unlike a cycloid on bicycle, which always moves forward)

• What is the sum of exterior angles of any convex polygon?
• Instructor Notes: Explain exterior angles through the first diagram below

• Answer: 360 degrees. Basis the visualization in second figure above

• 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.
• Several marbles are placed on circular track of circumference 1m. At time zero, each marble starts to move on the track at speed of 1 m/minute, randomly either clockwise or anticlockwise. All collisions result in marbles bouncing back with no change in speed. Where will the marbles be at end of 1 minute, relative to their original positions?
• Answer: Same position. When collisions happen, we can just see that as marbles crossing over each other.
• 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

• 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.