Batch 2 - Class 297 - Hopper Games

• (Expii) After a deep hundred year slumber, Ant the ant wakes up in an unfamiliar, large, dark square forest. Ant is unaware of its exact location or direction, but it does know that its starting point is exactly 10 meters away from the nearest straight edge. What is the shortest distance Ant must travel to guarantee escape from the forest? Round your answer, in meters, to the nearest hundredth.
• One potential is for the ant to go in any one direction for 10m, and then go around the 10m radius circle around the original wake-up point (so, 10 + 2*pi*10). Is there a better answer?
• Answer: https://math.stackexchange.com/questions/604824/can-this-ant-find-its-way-back-to-the-nest - the intuition is to walk outside the circle a bit, then take a tangent from that point back to the circle, go around the circle and at the end again get on a tangent.

• Revise the notion of probability and tree diagrams with kids. Take a sample tree diagram of a coin toss, two coin tosses and so on
• "Is it Fair" activity: Put a red-red and red-blue chip in a cup. Shake the cup and drop the chips. Player A wins if the chip are red-red, and Player B wins if the chips are red-blue. Best of 10 wins the game.
• Do you think the game is fair? Why? Make a prediction. Now play ten games and tabulate your results.
• Basis these trials, do you think the game is fair?
• Draw the tree diagram. Calculate the theoretical chances of Player A winning versus Player B.
• Now add another red-red chip. Player A wins with red-red-red, Player B wins with red-red-blue. Is this game fair? Play the game and figure out. 
• Suppose we use a red-red and two red-blue chips. How does that change outcomes? Is the game fair? Draw a tree diagram to analyze the results.
• Now lets use a red-white, red-blue, white-blue chip. Player A wins if all three chips fall with different colors. Player B scores if two chips match. Is this game fair? Why? Play and observe. Draw a tree diagram and calculate the theoretical chances.
• If the game is not fair, suggest a way of making it fair.
• "A Fair Hopper" activity: The game is played on a board as under. The chip starts at H. Now toss a coin thrice. For each toss, move the coin to the right if its heads, and to the left if it is tails. After three tosses, Player A scores a point if the chip is on either I square. Player B scores a point if the chip is on H, J or K squares.

• Predict who will win the game - is the game fair? Play and observe for 10 games. Use a tree diagram to analyze the game and determine the fairness of the game. If this game is not fair, how could you change to make it fair?
• Suppose we toss the coin four times, and add another "L" slot on either side of board. Player A now wins at I & L, and Player B wins at all other locations. Is the game fair now? Play and observe. Draw a tree diagram and determine who has better chances of winning. Can we make this game fair? 
• What do you think if we use an even longer board?
• "Can a Happy Hopper Escape?" Imagine a hopper stranded on a 3x3 square island with two bridges labeled B. Use a chip (a "happy hopper") with an arrow marked on top. Initially, the happy hopper is at the center of island (H) with arrow pointing up. For each turn, toss the coin twice. On the first, rotate the hopper 90 degree counterclockwise if its a heads, and clockwise if tails. For the second toss, move the hopper in direction of arrow for a head, and opposite to arrow for a tail (without rotating the hopper). Play till the hopper lands on the bridge or falls in ocean.

• Does the happy hopper have a fair chance to escape? Make a prediction. Play the game 10 times - record the sequence of squares (you may mark the squares as shown in right image above). Does the game appear fair?
• Note that this game has a recursive thinking, and may be useful to think of this in sets of 4 tosses each, as one analyzes all possibilities. The probability of landing on the bridge is roughly 1/6. 
• Suppose we attach two more bridges at (2,0) and (0,2) - does that improve the chances of escape?
• If we had to add another bridge to make the game fair, where would you add it?
• In "Happy Hopper Escape" game above, suppose you had a 5x5 island with two bridges. How does that change the chances of escape? How could you put a set of bridges on a 5x5 island to make it fair?

• Design your own game on lines of Hopper. Analyze the fairness of the game.