CAT 2025 Slot 2 · DILR · Set 3: Balls and Hoops
The full set and all 4 questions (Q10–Q13) from CAT 2025 Slot 2, with answers and solutions.
There are six spherical balls, B1, B2, B3, B4, B5, and B6, and four circular hoops H1, H2, H3, and H4.
Each ball was tested on each hoop once, by attempting to pass the ball through the hoop. If the diameter of a ball is not larger than the diameter of the hoop, the ball passes through the hoop and makes a “ping”. Any ball having a diameter larger than that of the hoop gets stuck on that hoop and does not make a ping.
The following additional information is known:
1. B1 and B6 each made a ping on H4, but B5 did not.
2. B4 made a ping on H3, but B1 did not.
3. All balls, except B3, made pings on H1.
4. None of the balls, except B2, made a ping on H2.
Question 10 · Games and Tournaments
What was the total number of pings made by B1, B2, and B3?
Answer & solution
Answer: 6
A ping on a hoop means ball ≤ hoop; no ping means ball > hoop.
Fact 4: B2 ≤ H2, and every other ball > H2.
Fact 2: B4 ≤ H3 < B1.
Fact 1: B1 ≤ H4 < B5 (and B6 ≤ H4).
Fact 3: B5 ≤ H1 < B3.
Chaining: B2 ≤ H2 < B4 ≤ H3 < B1 ≤ H4 < B5 ≤ H1 < B3.
B3 is larger than H1, the largest hoop, so it makes 0 pings. B2 fits H2, the smallest hoop, so it pings all 4. B1 is larger than H2 and H3 but fits H4 and H1, so it makes 2 pings.
Total = 2 + 4 + 0 = 6.
Shortcut: B3 is the biggest ball (no pings) and B2 the smallest (four pings). B1 is blocked at H3 but passes H4, so only the two largest hoops.
Hence, the answer is 6.
Question 11 · Games and Tournaments
Which of the following statements about the relative sizes of the balls is NOT NECESSARILY true?
- A
B4 < B5 < B3
- B
B2 < B1 < B5
- C
B1 < B6 < B3
- D
B1 < B5 < B3
Answer & solution
Answer: Option C
The facts chain into B2 ≤ H2 < B4 ≤ H3 < B1 ≤ H4 < B5 ≤ H1 < B3.
B6 is placed only loosely: it pings H4 (B6 ≤ H4) and does not ping H2 (B6 > H2). It is never compared with H3, B4 or B1.
Options 1, 2 and 4 follow directly from the chain.
Option 3 needs B1 < B6. But B6 could be small enough to pass H3, making it smaller than B1. So it is not necessarily true.
Shortcut: find the ball the clues leave loose (B6, pinned only between H2 and H4). Any option that compares it with B1 or B4 cannot be forced.
Hence, the answer is option 3.
Question 12 · Games and Tournaments
Which of the following statements about the relative sizes of the hoops is true?
- A
H2 < H3 < H4 < H1
- B
H1 < H3 < H4 < H2
- C
H1 < H4 < H3 < H2
- D
H2 < H4 < H3 < H1
Answer & solution
Answer: Option A
B4 pings H3 but not H2, so H2 < B4 ≤ H3, giving H2 < H3.
B1 pings H4 but not H3, so H3 < B1 ≤ H4, giving H3 < H4.
B5 pings H1 but not H4, so H4 < B5 ≤ H1, giving H4 < H1.
Together: H2 < H3 < H4 < H1.
Shortcut: each 'pings hoop X but not hoop Y' clue tells you Y < X. Three such clues order all four hoops.
Hence, the answer is option 1.
Question 13 · Games and Tournaments
What BEST can be said about the total number of pings from all the tests undertaken?
- A
12 or 13
- B
13 or 14
- C
12 or 13 or 14
- D
At least 9
Answer & solution
Answer: Option A
Hoops in increasing size: H2 < H3 < H4 < H1, with B2 ≤ H2 < B4 ≤ H3 < B1 ≤ H4 < B5 ≤ H1 < B3.
B2 fits all four hoops: 4 pings.
B4 fails H2, fits H3, H4, H1: 3 pings.
B1 fits only H4 and H1: 2 pings.
B5 fits only H1: 1 ping.
B3 fits none: 0 pings.
B6 fails H2 and fits H4 and H1; whether it fits H3 is unknown: 2 or 3 pings.
Total = 4 + 3 + 2 + 1 + 0 + (2 or 3) = 12 or 13.
Shortcut: five balls are fixed at 4, 3, 2, 1 and 0 pings (sum 10); only B6 varies, between 2 and 3.
Hence, the answer is option 1.
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