Batch 3 - Class 240 - The Game of Criss-Cross

• (Mott - 32) Mysterious Computation. Professor Digit claims that 161+134+145=503. Further, he also claims that 2625 when divided by 52 gives a quotient of 36 and zero remainder. Is he nuts?
• Answer: The computations are being made in base 7 instead of base 10

• Let kids play this game and figure out a winning strategy. Split up kids into groups of two, and let them play against each other.
• Instead of seven points, ask them to change the number of points and see who wins.
• Instead of taking a triangle as a starting point, ask them to take a square, or a pentagon.
• Ask them to count the number of vertices, edges, and faces at the end of the game and tabulate them
• What relationships do they observe?
• Can they use these patterns to predict who will win the game?
• Students should come to a conclusion that the winner only depends on number of points, and not the shape or how you place the points
• Are they able to arrive at the relationship between number of faces and edges? Why does that 2E=3F hold? Notice that this holds only when all possible edges have been drawn, and not at each stage.
• Can they see the relationship V-E+F=2? If not, start from one vertex to illustrate the base case (E=0, F=1), and then add a vertex - what edges must/can be added for the graph to be connected, and when do you start to add faces and edges in tandem.
• With both these relationships, can you deduce the number of edges possible, and hence who should win the game?

• If the square around the four vertices was already formed and all edges had to be inside it, what would be the result of the game?
• Opposite, because the one edge we were drawing outside the square can't be drawn now. Now 3F+1=2E
• What if you did the same thing on a pentagonal board (i.e. now edges can go outside the boundary)
• All planar graphs (drawn on a plane, no edges crossing) have V-E+F=2
• Think about a cube. What is V-E+F for that? However, a cube is not a planar graph. Can you think about how to convert the cube to a planar graph - what's the parallel?
• Can you think about how to convert any polyhedron to a planar graph? Think about a prism for example.

• A real puzzle with truth tables - Bob, Ted,and Carol play a hat game. This particular game is played with 5 hats, three red, and two blue. Each person gets one hat put on his or her head. Bob can see Ted’s hat and Carol’s hat but can’t see his own. Ted can see Bob’s hat and Carol’s hat, but not his own. Carol cannot see any of the hats. The game is to see which player is the first to figure out what color hat he or she is wearing. Bob goes first. He says he can’t tell. Then Ted says he can’t tell. Carol now says she is wearing a red hat. How did she know?
• Draw the equivalent of truth table for this (instead of T and F, it will be color of hats - first draw all possibilities, then start to reduce possibilities)
• "Constraining the truth table" can provide answers
• Answer: Notice that in the final analysis, all four possibilities have Carol wearing the Red Hat