In a certain game, a player can accumulate points only by scoring either an X, which counts 3 points, or...

1. Translate the problem requirements: We need to find how many different values 't' (points from X's worth 3 points each) can take when the total score is 24 points, with remaining points coming from Y's worth 2 points each.
2. Set up the constraint equation: Express the total points as a combination of X's and Y's, then determine what values of t are mathematically possible.
3. Apply the divisibility condition: Since Y's give 2 points each, the remaining points (24-t) must be divisible by 2, which constrains the possible values of t.
4. Test boundary conditions: Verify that each possible value of t corresponds to a realistic scenario where both the number of X's and Y's are non-negative integers.
5. Count the valid solutions: Enumerate all values of t that satisfy both the mathematical constraints and the real-world conditions of the game.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this game. A player can score in two ways:

• Score an X = get 3 points
• Score a Y = get 2 points

The player ended up with 24 points total. We're told that 't' represents the points that came from scoring X's. This means the remaining points (24 - t) must have come from scoring Y's.

What we need to find: How many different values can 't' have?

Process Skill: TRANSLATE - Converting the word problem into mathematical relationships

2. Set up the constraint equation

Let's think about this step by step. If 't' points came from X's, then:

• Number of X's scored = \(\mathrm{t} ÷ 3\) (since each X gives 3 points)
• Points from Y's = \(24 - \mathrm{t}\)
• Number of Y's scored = \((24 - \mathrm{t}) ÷ 2\) (since each Y gives 2 points)

For this to make sense in the real game, both the number of X's and number of Y's must be whole numbers (you can't score half an X or half a Y).

So we need:

• t must be divisible by 3 (so we get a whole number of X's)
• (24 - t) must be divisible by 2 (so we get a whole number of Y's)

3. Apply the divisibility condition

Let's work with the condition that \((24 - \mathrm{t})\) must be divisible by 2.

Since 24 is even, \((24 - \mathrm{t})\) will be even when t is even, and odd when t is odd.
For \((24 - \mathrm{t})\) to be divisible by 2, we need \((24 - \mathrm{t})\) to be even.
Therefore, t must be even.

Now we need t to satisfy both conditions:

• t must be divisible by 3
• t must be even

This means t must be divisible by both 2 and 3, so t must be divisible by 6.

Possible values of t: 0, 6, 12, 18, 24, 30, ...

4. Test boundary conditions

Now let's check which values actually work within our constraints. Remember, we need \(0 ≤ \mathrm{t} ≤ 24\) (you can't score negative points or more than the total).

Let's test each possibility:

• t = 0: 0 X's, \((24-0)÷2 = 12\) Y's ✓
• t = 6: \(6÷3 = 2\) X's, \((24-6)÷2 = 9\) Y's ✓
• t = 12: \(12÷3 = 4\) X's, \((24-12)÷2 = 6\) Y's ✓
• t = 18: \(18÷3 = 6\) X's, \((24-18)÷2 = 3\) Y's ✓
• t = 24: \(24÷3 = 8\) X's, \((24-24)÷2 = 0\) Y's ✓

All of these give us non-negative whole numbers for both X's and Y's.

Process Skill: CONSIDER ALL CASES - Systematically checking each possible scenario

5. Count the valid solutions

Let's list all the valid values of t:

• t = 0 (0 X's, 12 Y's)
• t = 6 (2 X's, 9 Y's)
• t = 12 (4 X's, 6 Y's)
• t = 18 (6 X's, 3 Y's)
• t = 24 (8 X's, 0 Y's)

Counting these up: 0, 6, 12, 18, 24 → That's 5 different values.

Final Answer

The value t can take 5 different values: 0, 6, 12, 18, and 24.

The answer is B. Five.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what 't' represents

Students often confuse 't' as representing the number of X's scored rather than the total points from X's. This leads them to set up incorrect equations where they think t = number of X's instead of understanding that t = total points from X's, which means number of X's = \(\mathrm{t}÷3\).

2. Setting up the wrong constraint equation

Many students incorrectly write the main equation as \(3\mathrm{x} + 2\mathrm{y} = 24\) (where x and y are numbers of each score type) and then try to find values of x, missing the connection that \(\mathrm{t} = 3\mathrm{x}\). They fail to recognize that the question is asking for possible values of t (points from X's), not the number of ways to score.

3. Missing the divisibility requirements

Students often overlook that both the number of X's and Y's must be whole numbers, which creates the crucial constraints that t must be divisible by 3 and (24-t) must be divisible by 2. Without recognizing these constraints, they might consider any value of t between 0 and 24.

Errors while executing the approach

1. Incorrectly applying the divisibility by 6 rule

Even when students correctly identify that t must be divisible by both 2 and 3, they may make errors in finding multiples of 6. Some students might miss that 0 is divisible by 6, or incorrectly include values like 30 without checking if they exceed the maximum possible score of 24.

2. Arithmetic errors in verification

When testing each possible value of t, students frequently make calculation mistakes such as incorrectly computing \((24-\mathrm{t})÷2\) or \(\mathrm{t}÷3\). For example, when t=18, they might calculate \((24-18)÷2 = 4\) instead of 3, leading them to incorrect conclusions about which values are valid.

Errors while selecting the answer

1. Counting error in final enumeration

After correctly finding all valid values (0, 6, 12, 18, 24), students may miscount them. They might count 4 values by missing one (often forgetting that 0 is a valid case), or count 6 values by double-counting or including an invalid value they didn't properly eliminate.