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 balls is NOT NECESSARILY true?

The correct option is: B1 < B6 < B3

To solve this logical reasoning problem, let's denote the diameter of each ball by its corresponding name (e.g., $B_1$, $B_2$, etc.) and the diameter of each hoop as $H_1$, $H_2$, etc. According to the problem's rules, if a ball makes a "ping," it means its diameter is less than or equal to the hoop's diameter.

$B_i\le H_j$

Conversely, if a ball gets stuck and does not make a ping, its diameter must be strictly greater than the hoop's diameter.

$B_i>H_j$

Let's break down the information step-by-step to determine the relative sizes of all the balls:

Step 1: Analyze statement 4
"None of the balls, except B2, made a ping on H2."
This tells us that ball B2 passed through hoop H2, but all other balls were too large. Therefore, B2 must be the ball with the strictly smallest diameter among all the balls.

Step 2: Analyze statement 3
"All balls, except B3, made pings on H1."
This implies that all balls except B3 could pass through hoop H1, meaning B3 was the only ball too large for it. Consequently, B3 must be the ball with the strictly largest diameter among all the balls.

Step 3: Analyze statement 2
"B4 made a ping on H3, but B1 did not."
This means the diameter of B4 is less than or equal to the diameter of H3, while the diameter of B1 is strictly greater than H3. By combining these facts, we can deduce that B4 is smaller than B1.

$B_4<B_1$

Step 4: Analyze statement 1
"B1 and B6 each made a ping on H4, but B5 did not."
This means the diameters of both B1 and B6 are less than or equal to the diameter of H4, but the diameter of B5 is strictly greater than H4. This allows us to establish two inequalities:

$B_1<B_5$

$B_6<B_5$

Step 5: Combine the findings
By chaining together the inequalities from our deductions, we can establish a clear size order for most of the balls. Knowing B2 is the smallest, B3 is the largest, B4 < B1, and B1 < B5, we get the following definite sequence:

$B_2<B_4<B_1<B_5<B_3$

We also know that $B_6<B_5$, $B_2<B_6$, and $B_6<B_3$. However, the exact position of B6 relative to B4 and B1 remains unknown based on the given information.

Step 6: Evaluate the options
Let's check each option against our established size sequence:

Option: B4 < B5 < B3
This is necessarily true because $B_4<B_1<B_5<B_3$.

Option: B2 < B1 < B5
This is necessarily true because $B_2<B_4<B_1<B_5$.

Option: B1 < B5 < B3
This is necessarily true because $B_1<B_5<B_3$.

Option: B1 < B6 < B3
This statement claims that B1 must be smaller than B6. While we know both B1 and B6 are smaller than B5, the problem provides no information to determine the relative size between B1 and B6. It is entirely possible that B6 is smaller than B1. Therefore, this statement is NOT NECESSARILY true.