Question 10 - Figure it out - Page 72, 73 - Chapter 3 Class 6 - Number Play (Ganita Prakash)

Transcript

Question 10 Starting with 0, players alternate adding numbers between 1 and 3. The first person to reach 22 wins. What is the winning strategy now?The Rules of the Game This is a counting game for two players with the following rules: The game starts at a total of 0. Two players take turns. On each turn, a player can add 1, 2, or 3 to the current total. The first player to say the number 22 wins. Example of a Game Here is a sample game where Player 1 knows the winning strategy and Player 2 does not. Turn 1: Player 1 starts and adds 2. New Total: 2 Turn 2: Player 2 adds 3. New Total: 5 Turn 3: Player 1 adds 1. New Total: 6 Turn 4: Player 2 adds 3. New Total: 9 Turn 5: Player 1 adds 1. New Total: 10 Turn 6: Player 2 adds 2. New Total: 12 Turn 7: Player 1 adds 2. New Total: 14 Turn 8: Player 2 adds 3. New Total: 17 Turn 9: Player 1 adds 1. New Total: 18 Turn 10: Player 2 adds 3. New Total: 21 Turn 11: Player 1 adds 1. New Total: 22 Player 1 wins! Now, let's deduce the strategy they used. The Winning Strategy The strategy is to identify a sequence of "safe" numbers that you must say to guarantee a win. We can find these numbers by working backward from the goal. The Goal: The winning number is 22. The Key Interval: A player can add 1, 2, or 3. The key to controlling the game is to think about the jump from one of your turns to the next. No matter what your opponent adds (1, 2, or 3), you can always make the total jump between your turns equal to 4. If they add 1, you add 3 (1+3=4). If they add 2, you add 2 (2+2=4). If they add 3, you add 1 (3+1=4). Working Backward: Since you can control the game in steps of 4, the winning numbers are found by subtracting 4 from the goal. To guarantee you say 22, you must first say 22 - 4 = 18. To guarantee you say 18, you must first say 18 - 4 = 14. To guarantee you say 14, you must first say 14 - 4 = 10. To guarantee you say 10, you must first say 10 - 4 = 6. To guarantee you say 6, you must first say 6 - 4 = 2. The Winning Sequence The Winning Sequence of numbers to say is: 2, 6, 10, 14, 18, 22. The first player can always win this game. On their first turn, they should add 2. After that, they just need to make sure their move always lands on the next number in the winning sequence.