Loading...
Rita and Sam play the following game with \(\mathrm{n}\) sticks on a table. Each must remove \(1, 2, 3, 4\text{ or }5\) sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of \(\mathrm{n}\) such that Sam can always win no matter how Rita plays?
Let's break down what this game is all about in simple terms:
Think of this like a strategic game where the key is not just playing well, but having a foolproof plan that works against any possible moves your opponent makes.
Process Skill: TRANSLATE - Converting the game rules into clear strategic requirements
Let's work backwards from the end to understand when someone is guaranteed to win or lose:
If it's your turn and there are 1-5 sticks left: You win! You can take all remaining sticks.
If it's your turn and there are exactly 6 sticks left: You're in trouble! No matter what you do:
So 6 sticks is a "losing position" - whoever faces 6 sticks on their turn will lose against perfect play.
If it's your turn and there are 7-11 sticks: You can win by forcing your opponent into the losing position of 6:
Continuing this logic:
If it's your turn and there are exactly 12 sticks: You're in trouble again!
The pattern emerges: The losing positions are multiples of 6!
Why this pattern works:
If you're facing a multiple of 6 sticks, any move you make (removing 1-5 sticks) will leave your opponent with a number that is NOT a multiple of 6. From any non-multiple of 6, your opponent can always remove the right number of sticks to get back to a multiple of 6, putting you in a losing position again.
Process Skill: INFER - Recognizing the non-obvious pattern that multiples of 6 create losing positions
Now we apply our discovery:
For Sam to guarantee a win, Rita must start her first turn facing a losing position. This means we need a starting number that is a multiple of 6.
Looking at our answer choices:
Verification with n = 12:
Rita faces 12 sticks on her first turn. No matter what she takes:
Sam can always respond to put Rita back in a losing position!
The answer is D. 12
With 12 sticks, Sam has a guaranteed winning strategy: always respond to Rita's moves by leaving her with a multiple of 6 sticks. Since Rita goes first and starts with 12 sticks (a multiple of 6), she begins in a losing position and Sam can maintain his advantage throughout the game.
1. Misunderstanding who needs the winning strategy
Students often confuse which player needs to have the guaranteed winning strategy. The question asks for values where "Sam can always win," but students might look for positions where Rita (who goes first) has the advantage. This leads them to seek winning positions for the first player instead of losing positions.
2. Failing to work backwards from the end game
Many students try to work forward from the starting position instead of analyzing backwards from winning/losing end states. They might attempt to simulate games or look for patterns by testing small numbers without first establishing what constitutes a fundamental winning or losing position (like the critical "6 sticks" losing position).
3. Not recognizing this as a game theory problem requiring guaranteed strategies
Students may approach this as a probability or simulation problem rather than a strategic game theory problem. They might think about "likely" outcomes or try to find "good" strategies instead of seeking positions that guarantee victory regardless of the opponent's moves.
1. Incorrectly identifying the losing position pattern
Even when students understand they need to work backwards, they might incorrectly conclude that positions like 5, 7, or other numbers are the fundamental losing positions. This leads to wrong patterns (like multiples of 5 or 7) instead of recognizing that 6 is the key losing position.
2. Arithmetic errors in pattern recognition
When checking if numbers are multiples of 6, students may make division errors. For example, they might incorrectly think 10 or 16 are multiples of 6, or conversely, doubt that 12 is a multiple of 6. These basic arithmetic mistakes derail the entire solution.
3. Incomplete verification of the strategy
Students might identify that 12 is a multiple of 6 but fail to verify that Sam can actually maintain his advantage after Rita's first move. They may not check all possible responses (what happens when Rita takes 1, 2, 3, 4, or 5 sticks) to confirm that Sam can always force Rita back to a multiple of 6.
4. Errors while selecting the answer
No likely faltering points