Question

What was the total number of pings made by B1, B2, and B3?

• A. B4 < B5 < B3
• B. B2 < B1 < B5
• C. B1 < B5 < B3
• D. B1 < B6 < B3
• A. H1 < H4 < H3 < H2
• B. H2 < H3 < H4 < H1
• C. H1 < H3 < H4 < H2
• D. H2 < H4 < H3 < H1
• A. 13 or 14
• B. At least 9
• C. 12 or 13
• D. 12 or 13 or 14

Answer 1

Correct Answer: 6

Step 1: Interpret the conditions and establish size relations. A ping occurs whenever the diameter of a ball is less than or equal to the diameter of a hoop. Using the given rules, we determine the relative sizes of the balls and hoops. From Rule 4, ball \(B2\) fits into hoop \(H2\), while all other balls are larger than \(H2\). Hence, \(B2\) is the smallest ball. From Rule 3, ball \(B3\) is larger than \(H1\), whereas all other balls are less than or equal to \(H1\). Thus, \(B3\) is the largest ball. Rule 2 implies \(B1 > H3\) and \(B4 \le H3\), giving \(B1 > B4\). Rule 1 states \(B5 > H4\), while \(B1 \le H4\) and \(B6 \le H4\), which implies \(B5 > B1\) and \(B5 > B6\). Combining these deductions, the balls are ordered as \[ B3 > B5 > B1 > B4 > B2, \] and the hoops are ordered as \[ H1 > H4 > H3 > H2. \] Step 2: Count pings for each required ball. For ball \(B1\): \[ B1 \le H1 \Rightarrow \text{ping}, \quad B1 > H2 \Rightarrow \text{no ping}, \] \[ B1 > H3 \Rightarrow \text{no ping}, \quad B1 \le H4 \Rightarrow \text{ping}. \] Hence, \(B1\) produces \(2\) pings. For ball \(B2\): Since \(B2\) is smaller than \(H2\), the smallest hoop, it must be smaller than all hoops. Thus, \(B2\) pings on \(H1, H2, H3,\) and \(H4\). Total pings for \(B2 = 4\). For ball \(B3\): Since \(B3\) is larger than \(H1\), the largest hoop, it cannot fit into any hoop. Therefore, \(B3\) produces \(0\) pings. Step 3: Compute the total number of pings. Adding the pings from \(B1, B2,\) and \(B3\), \[ \text{Total pings} = 2 + 4 + 0 = 6. \] Hence, the total number of pings for balls \(B1, B2,\) and \(B3\) is \(6\).

Answer 2

Step 1: Understanding the Question and Initial Deductions:
We need to find the total number of pings for balls B1, B2, and B3 across all four hoops. A ping occurs if Diameter(Ball) $\le$ Diameter(Hoop). Let's use Bx and Hx to denote the diameters.
From the rules, we can deduce the relative sizes.
- Rule 4 implies B2 $\le$ H2 and (B1, B3, B4, B5, B6) > H2. This makes B2 the smallest ball.
- Rule 3 implies B3 > H1 and (B1, B2, B4, B5, B6) $\le$ H1. This makes B3 the largest ball.
- From Rule 2, B1 > H3 and B4 $\le$ H3. This implies B1 > B4.
- From Rule 1, B5 > H4, while B1 $\le$ H4 and B6 $\le$ H4. This implies B5 > B1 and B5 > B6.
- Combining these, we get a partial order for balls: B3 > B5 > B1 > B4 > B2.
- For hoops: H1 $\ge$ B5 > H4 $\ge$ B1 > H3 $\ge$ B4 > H2 $\ge$ B2. This gives a clear order: H1 > H4 > H3 > H2.
Step 2: Calculating Pings for Each Ball:
- Pings for B1:
- vs H1: B1 $\le$ H1 (Rule 3) $\implies$ Ping.
- vs H2: B1 > H2 (Rule 4) $\implies$ No Ping.
- vs H3: B1 > H3 (Rule 2) $\implies$ No Ping.
- vs H4: B1 $\le$ H4 (Rule 1) $\implies$ Ping.
- Total for B1 = 2 pings.
- Pings for B2:
- B2 is the smallest ball. It is smaller than H2, which is the smallest hoop. Therefore, B2 is smaller than all hoops.
- vs H1: B2 $\le$ H1 (Rule 3) $\implies$ Ping.
- vs H2: B2 $\le$ H2 (Rule 4) $\implies$ Ping.
- vs H3: Since H3 > H2 $\ge$ B2, B2 $\le$ H3 $\implies$ Ping.
- vs H4: Since H4 > H2 $\ge$ B2, B2 $\le$ H4 $\implies$ Ping.
- Total for B2 = 4 pings.
- Pings for B3:
- B3 is the largest ball. It is larger than H1, which is the largest hoop. Therefore, B3 is larger than all hoops.
- vs H1: B3 > H1 (Rule 3) $\implies$ No Ping.
- It will not ping on any other smaller hoop either.
- Total for B3 = 0 pings.
Step 3: Final Answer:
Total number of pings for B1, B2, and B3 is the sum of their individual pings.
Total = Pings(B1) + Pings(B2) + Pings(B3) = 2 + 4 + 0 = 6.

Answer 3

To determine which statement about the relative sizes of the balls is NOT NECESSARILY true, we need to analyze the given conditions and test each option.

Step-by-Step Analysis:

1. Understand the conditions provided in the comprehension:
   • B1 and B6 made a ping on H4, meaning their diameters are not larger than the diameter of H4. B5 did not make a ping on H4, indicating B5's diameter is larger than H4.
   • B4 made a ping on H3, but B1 did not, so B1's diameter is larger than H3, while B4's diameter is not larger.
   • All balls except B3 made a ping on H1, implying B3's diameter is larger than H1, while others are not larger.
   • None of the balls except B2 made a ping on H2, indicating B2's diameter is not larger than H2, while others are larger.
2. Analyze the options:
   • Option 1: B4 < B5 < B3
     • B4 made a ping on H3, whereas B1 did not, so B4 could be less than B1.
     • B5 did not make a ping on H4, whereas B1 and B6 did, suggesting B5 is larger than B1 and B6; however, we don't have explicit information on B5 concerning B4 or B3.
     • Option B4 < B5 < B3 remains plausible based on available data.
   • Option 2: B2 < B1 < B5
     • B2 made a ping on H2, while others did not, meaning B2 is the smallest.
     • B1 did not make a ping on H3, indicating B1 is not the smallest and likely < B5, as B5 did not make a ping on H4.
     • This option consistently aligns with known facts.
   • Option 3: B1 < B5 < B3
     • B5 did not make a ping on H4, nor did B1 on H3; this makes B5 plausibly larger and able to fit between B1 and B3.
     • B3 did not make a ping on H1, suggesting it could be larger, fitting the order.
     • This option aligns with known facts.
   • Option 4: B1 < B6 < B3
     • B1 did not make a ping on H3, and B6 made a ping on H4, possibly smaller than B3.
     • B3 did not make a ping on H1, possibly larger than B6.
     • However, the statement B1 < B6 is not conclusively supported by the given conditions, specifically concerning their performances on other hoops where no direct comparison can be validated.
     • Thus, this option is less certain and is the one NOT NECESSARILY true.

Conclusion:

The statement B1 < B6 < B3 is NOT NECESSARILY true based on the provided data, as the exact sizes of B1 and B6 relative to each other cannot be determined from the known conditions. Hence, the correct answer is:

B1 < B6 < B3

 

Answer 4

Step 1: Understanding the Question:
We need to identify which of the given inequalities about the ball sizes cannot be definitively proven from the information given. We will use the size relationships derived in the previous question. 
Step 2: Reviewing Ball Size Deductions: 
- B3 is the largest ball.
- B2 is the smallest ball.
- B5 > H4 and B1 $\le$ H4 $\implies$ B5 > B1.
- B1 > H3 and B4 $\le$ H3 $\implies$ B1 > B4.
- B5 > H4 and B6 $\le$ H4 $\implies$ B5 > B6.
- The definite order is: B3 > B5 > B1 > B4 > B2.
- The position of B6 is uncertain. We only know B5 > B6 and B6 > H2 > B2. The relationship between B6 and B1, and B6 and B4 is not determined by the given rules. 
Step 3: Evaluating the Options: 
- (A) B4 < B5 < B3:
- Is B5 < B3? Yes, B3 is the largest.
- Is B4 < B5? Yes, we established B5 > B1 and B1 > B4, so B5 > B4.
- This statement is necessarily true. 
- (B) B2 < B1 < B5:
- Is B1 < B5? Yes, we established this.
- Is B2 < B1? Yes, B2 is the smallest.
- This statement is necessarily true. 
- (C) B1 < B5 < B3:
- Is B5 < B3? Yes.
- Is B1 < B5? Yes.
- This statement is necessarily true. 
- (D) B1 < B6 < B3:
- Is B6 < B3? Yes, B3 is the largest.
- Is B1 < B6? This is unknown. From Rule 1, we have B1 $\le$ H4 and B6 $\le$ H4. This does not allow us to compare B1 and B6. It is possible that B1 < B6, B1 > B6, or B1 = B6.
- Since we cannot prove B1 < B6, the entire statement is not necessarily true. 
Step 4: Final Answer: 
The relationship between B1 and B6 cannot be determined from the given information. Therefore, the statement "B1 < B6 < B3" is not necessarily true. 
 

Answer 5

To solve the problem, we need to determine the relative sizes of the hoops H1, H2, H3, and H4 based on the given conditions. We have information about which balls made a ping when tested with each hoop, which indicates that the ball passed through the hoop.

1. Condition 1: B1 and B6 made a ping on H4, but B5 did not. 
   This implies the diameter of H4 is larger than or equal to the diameter of B1 and B6, but smaller than B5.
2. Condition 2: B4 made a ping on H3, but B1 did not. 
   This implies the diameter of H3 is larger than or equal to the diameter of B4, but smaller than B1.
3. Condition 3: All balls, except B3, made pings on H1. 
   This suggests that the diameter of H1 is larger than or equal to all other balls except B3. This makes H1 the largest hoop.
4. Condition 4: None of the balls, except B2, made a ping on H2. 
   This suggests that the diameter of H2 is the smallest as only B2 can pass through it.

Putting these conditions together, we need to order the hoops from smallest to largest:

• Since only B2 made a ping on H2, H2 is the smallest.
• H3 is larger than B4 but smaller than B1.
• H4 is larger than or equal to B1 and B6, but smaller than B5.
• H1 is larger than all others except B3; hence, it is the largest.

Therefore, the order of hoops from smallest to largest is: 
H2 < H3 < H4 < H1

Thus, the correct answer is:

H2 < H3 < H4 < H1

Answer 6

Step 1: Understanding the Question:
We need to find the correct ascending or descending order of the hoop sizes based on the deductions from the problem statement.
Step 2: Establishing the Hoop Order:
We will use the relationships between balls and hoops to determine the relative sizes of the hoops.
- From Rule 3, B3 > H1 and B5 $\le$ H1. From Rule 1, B5 > H4.
- Combining these: H1 $\ge$ B5 > H4. Therefore, H1 > H4.
- From Rule 1, B1 $\le$ H4. From Rule 2, B1 > H3.
- Combining these: H4 $\ge$ B1 > H3. Therefore, H4 > H3.
- From Rule 2, B4 $\le$ H3. From Rule 4, B4 > H2.
- Combining these: H3 $\ge$ B4 > H2. Therefore, H3 > H2.
Step 3: Combining the Inequalities and Final Answer:
Putting all the derived inequalities together, we get:
H1 > H4 > H3 > H2.
This can be written in ascending order as:
H2 < H3 < H4 < H1.
This matches option (B).

Answer 7

Step 1: Identify the task. There are 6 balls and 4 hoops, giving a total of \(6 \times 4 = 24\) tests. The objective is to determine the total number of pings across all these tests. Depending on the uncertainty in relative sizes, the answer may be a fixed value or a small range. Step 2: Determine pings for each ball. From the given size orders, \[ B3 > B5 > B1 > B4 > B2, \quad H1 > H4 > H3 > H2, \] and the additional information \[ B5 > B6 > B2. \] Analyzing each ball: - Ball \(B1\): It pings on \(H1\) and \(H4\), but not on \(H2\) or \(H3\). Total pings \(= 2\). - Ball \(B2\): Being the smallest ball, it pings on all four hoops. Total pings \(= 4\). - Ball \(B3\): Being the largest ball, it does not ping on any hoop. Total pings \(= 0\). - Ball \(B4\): It pings on \(H1\), does not ping on \(H2\), pings on \(H3\), and pings on \(H4\). Total pings \(= 3\). - Ball \(B5\): It pings on \(H1\) only, and does not ping on \(H2\), \(H3\), or \(H4\). Total pings \(= 1\). - Ball \(B6\): It pings on \(H1\), does not ping on \(H2\), and pings on \(H4\). Its interaction with \(H3\) cannot be uniquely determined, since its size relative to \(H3\) is unknown. Hence, total pings for \(B6\) are either \(2\) or \(3\). Step 3: Add the total number of pings. Summing the contributions from all balls, \[ \text{Total pings} = 2 + 4 + 0 + 3 + 1 + (2 \text{ or } 3). \] This gives \[ \text{Total pings} = 12 \text{ or } 13. \] Step 4: Conclusion. The total number of pings across all 24 tests can be either \(12\) or \(13\), depending on the relative size of ball \(B6\) compared to hoop \(H3\).

Answer 8

Step 1: Understanding the Question: 
We need to find the total number of pings across all 24 tests (6 balls x 4 hoops). The answer might be a specific number or a range if there is uncertainty. 
Step 2: Calculating Pings for Each Ball: 
We use the established size orders: B3 > B5 > B1 > B4 > B2 and H1 > H4 > H3 > H2. The position of B6 is B5 > B6 > B2. 
- B1 Pings: On H1(Y), H4(Y). On H2(N), H3(N). Total = 2.
- B2 Pings: Smallest ball, pings on all hoops. Total = 4.
- B3 Pings: Largest ball, pings on no hoops. Total = 0.
- B4 Pings:
- vs H1 (Y, since H1 is largest)
- vs H2 (N, rule 4)
- vs H3 (Y, rule 2)
- vs H4 (Y, since H4 > H3 $\ge$ B4)
- Total = 3.
- B5 Pings:
- vs H1 (Y, rule 3)
- vs H2 (N, rule 4)
- vs H3 (N, since B5 > B1 > H3)
- vs H4 (N, rule 1)
- Total = 1.
- B6 Pings:
- vs H1 (Y, rule 3)
- vs H2 (N, rule 4)
- vs H4 (Y, rule 1)
- vs H3: This is unknown. We know B1 > H3 and B4 $\le$ H3. We do not have information to place B6 relative to H3. B6 could be larger or smaller than H3.
- So, B6 pings on H3 if B6 $\le$ H3, and does not ping if B6 > H3. - Total = 2 or 3.
Step 3: Calculating the Total Number of Pings: 
Summing the pings for all balls: 
Total Pings = Pings(B1) + Pings(B2) + Pings(B3) + Pings(B4) + Pings(B5) + Pings(B6) 
Total Pings = 2 + 4 + 0 + 3 + 1 + (2 or 3) 
Total Pings = 10 + (2 or 3) 
- If B6 does not ping on H3, Total = 10 + 2 = 12.
- If B6 pings on H3, Total = 10 + 3 = 13.
Step 4: Final Answer: 
The total number of pings can be either 12 or 13, depending on the size of B6 relative to H3. The statement that best captures this is "12 or 13". This corresponds to option (C).
(Note: While "At least 9" is technically true, it is not the BEST description of the total, as we can prove the total must be at least 12. "12 or 13" is the most precise and accurate statement.)