Lesson 5
1. Lesson 5
1.5. Explore 3
Module 2: Logic and Geometry
Winning a game or solving a puzzle requires a problem-solving approach. In earlier lessons, you identified patterns in games and puzzles and then developed a strategy to fit the pattern. The strategy may even have been developed using steps similar to the ones you identified for using reasoning to solve problems. Strategies can be adapted to come up with the best possible strategy for winning a game. The same approach of adapting strategies can be applied whether you are trying to solve a puzzle or figuring out how to maximize seating in a cafeteria.
Read “Example 1: Using reasoning to determine possible winning plays” on page 53 of your textbook. As you read, think about the strategies each player might use to win the game. What other way could Frank win the game on his turn? What strategy could Tara use to win on her next turn, if Frank does not win on his turn?
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Analyzing possible strategies using inductive and deductive reasoning can be helpful in winning a game. By determining what your moves need to be based on your opponent’s possible strategy, you can stay one move ahead of your opponent. In other words, you use reasoning to look ahead and try to figure out what your opponent might do next.
For instance, in the game tic-tac-toe, there are only a few moves for the entire game. You can deduce your next move from your opponent’s previous move and your knowledge of the point of the game. Part of your strategy is trying to figure out how to get three X’s in a row. The other part is trying to figure out how to stop your opponent from getting three O’s in a row. This requires looking ahead and adapting your strategy based on where you and your opponent could place your X’s and O’s.
Did You Know?
Noughts and crosses or three-in-a-row are other names for tic-tac-toe. These games may have been played as far back as 1300 BCE in Egypt. Grid games such as tic-tac-toe have been recorded in carvings from ancient Roman times.
—Adapted from: CANAVAN-MCGRATH ET AL. Principles of Mathematics 11 TR, © 2012 Nelson Education Limited. p. 39. Reproduced by permission.
In chess, players may use inductive reasoning to determine their strategy. A player may study the current placement of pieces on the board while thinking about the opponent’s play in previous games. Evidence is used to identify patterns. Based on these patterns, a player can make conjectures about his or her opponents’ intended moves and strategy.
“If . . . then” statements can be used to help determine an appropriate strategy. For instance, you might ask yourself, “If I do this, then what move could my opponent make?” You can do this for any number of possible scenarios. “If I do this instead, then what move could my opponent make?”
Read “Example 2: Using deductive and inductive reasoning to determine a winning strategy” on page 54 of your textbook. In this example, both inductive and deductive reasoning are involved in the strategy.