Batch 3 - Class 168 - Islands (Graphs)

• Big Challenge Puzzle - Note that there is no fixed point here. Thus the balancing has to happen up and down!

• What can be the approach to this? For one, kids can color land and water areas
• Let kids start with smaller problems - they can make their own graphs
• Can they find their patterns?
• Move dot outwards to take any path out. If the number of lines being crossed to get out is odd, the red dot is on land, else water
• Does it matter what kid of shape the outward path has?

• What do you expect the average of number of walls for all cells to be? 2? Less than 2? More than 2? Does it depend on the diagram?
• Lets try out some possible diagrams
• Can you see a pattern? (in averages?)
• Count the cells and walls total - do you see a pattern here?
• (Harder) Why? In a graph vertices - edges + regions = components. Here if we label each cell with number of walls in excess of 2, and join all the non-zero points, we get a graph. If we plug in the vertices and edges into above equation, we will get that the sum of labels is 2 (components - regions). However in all such graphs, components = regions +1, since components is bodies of land or water, and regions is all bodies of land or water surrounded by water or land - so every component except the outermost body of land. Hence the difference is 2.

• (Dudeney - 98) There is a defective scale, due to unequal length of two arms, which shows the same weight with eight packets on right and three cans on left, as well as with one packet on left and six cans on right. If the true weight of a can is one ounce, how much does the packet weigh?

• Answer: 8 x (weight of can) x (distance) = 3, and (weight of can) = 6 x (distance). This gives weight of can as 3/2. [Note that distance is really the ratio of two distances]