Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4text{...

1. Translate the problem requirements: We need to find a starting number of sticks where the second player (Sam) can guarantee a win regardless of the first player's (Rita's) moves, when each player can remove 1-5 sticks per turn
2. Identify the fundamental winning pattern: Work backwards from the end position to understand what constitutes a winning vs losing position for each player
3. Discover the strategic pattern: Determine the key numbers that create losing positions for the player whose turn it is
4. Apply the pattern to find Sam's guaranteed win: Use the discovered pattern to identify which starting number ensures Rita will always be in a losing position

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what this game is all about in simple terms:

• Rita and Sam take turns removing sticks from a pile
• On each turn, a player MUST remove between 1 and 5 sticks (they can choose how many)
• Rita goes first
• Whoever removes the last stick(s) wins
• We need to find a starting number where Sam can guarantee he wins, no matter what Rita does

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

2. Identify the fundamental winning pattern

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:

• Take 1 stick → leaves 5 for opponent (opponent wins)
• Take 2 sticks → leaves 4 for opponent (opponent wins)
• Take 3 sticks → leaves 3 for opponent (opponent wins)
• Take 4 sticks → leaves 2 for opponent (opponent wins)
• Take 5 sticks → leaves 1 for opponent (opponent wins)

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:

• With 7 sticks: take 1, leave 6 for opponent
• With 8 sticks: take 2, leave 6 for opponent
• With 9 sticks: take 3, leave 6 for opponent
• With 10 sticks: take 4, leave 6 for opponent
• With 11 sticks: take 5, leave 6 for opponent

3. Discover the strategic pattern

Continuing this logic:

If it's your turn and there are exactly 12 sticks: You're in trouble again!

• No matter how many you take (1-5), you'll leave your opponent with 7-11 sticks
• We just saw that 7-11 sticks are all winning positions for whoever faces them
• So your opponent can force you back to a losing position

The pattern emerges: The losing positions are multiples of 6!

• 6 sticks = losing position
• 12 sticks = losing position
• 18 sticks = losing position
• And so on...

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

4. Apply the pattern to find Sam's guaranteed win

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:

1. 7 = not a multiple of 6
2. 10 = not a multiple of 6
3. 11 = not a multiple of 6
4. 12 = \(6 \times 2\) = multiple of 6 ✓
5. 16 = not a multiple of 6

Verification with n = 12:
Rita faces 12 sticks on her first turn. No matter what she takes:

• Takes 1 → leaves 11 for Sam → Sam takes 5 → leaves 6 for Rita (losing position)
• Takes 2 → leaves 10 for Sam → Sam takes 4 → leaves 6 for Rita (losing position)
• Takes 3 → leaves 9 for Sam → Sam takes 3 → leaves 6 for Rita (losing position)
• Takes 4 → leaves 8 for Sam → Sam takes 2 → leaves 6 for Rita (losing position)
• Takes 5 → leaves 7 for Sam → Sam takes 1 → leaves 6 for Rita (losing position)

Sam can always respond to put Rita back in a losing position!

4. Final Answer

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.

Common Faltering Points

Errors while devising the approach

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.

Errors while executing the approach

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