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.

Which of the following statements about the relative sizes of the hoops is true?

The correct option is: H2 < H3 < H4 < H1

Let's break down the given information step-by-step to determine the relative sizes of the hoops. A ball will pass through and make a "ping" if its diameter is less than or equal to the hoop's diameter. If a ball gets stuck and does not make a ping, its diameter is strictly greater than the hoop's diameter.

Let's use the labels B1 to B6 to represent the diameters of the corresponding balls, and H1 to H4 to represent the diameters of the hoops.

Step 1: Analyze Statement 1
"B1 and B6 each made a ping on H4, but B5 did not."
This tells us that the diameters of B1 and B6 are less than or equal to H4, while the diameter of B5 is strictly greater than H4.

$B1\le H4$

$B6\le H4$

$B5>H4$

Step 2: Analyze Statement 2
"B4 made a ping on H3, but B1 did not."
This means B4's diameter is less than or equal to H3, and B1's diameter is strictly greater than H3.

$B4\le H3$

$B1>H3$

From Step 1, we know that $B1\le H4$. Combining this with $B1>H3$, we can form a logical chain of inequalities:

$H3<B1\le H4$

Therefore, it must be true that $H3<H4$.

Step 3: Analyze Statement 3
"All balls, except B3, made pings on H1."
This means the diameters of B1, B2, B4, B5, and B6 are all less than or equal to H1. Specifically, this means $B5\le H1$.

From Step 1, we know that $B5>H4$. Combining this with $B5\le H1$, we get another chain:

$H4<B5\le H1$

Therefore, it must be true that $H4<H1$.

Step 4: Analyze Statement 4
"None of the balls, except B2, made a ping on H2."
This means the diameters of B1, B3, B4, B5, and B6 are all strictly greater than H2. Specifically, this means $B4>H2$.

From Step 2, we know that $B4\le H3$. Combining this with $B4>H2$, we establish our final chain:

$H2<B4\le H3$

Therefore, it must be true that $H2<H3$.

Step 5: Final Conclusion
By chaining together the definitive hoop relationships determined in the previous steps, we have:

$H2<H3$
$H3<H4$
$H4<H1$

Combining all of these relationships gives us the final order of the hoop sizes from smallest to largest:

$H2<H3<H4<H1$

This confirms that the provided correct option is the only valid sequence based on the logical constraints.