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.

What BEST can be said about the total number of pings from all the tests undertaken?

The correct option is 12 or 13.

To solve this, we need to find the total number of "pings" made by all six balls (B1, B2, B3, B4, B5, B6) on all four hoops (H1, H2, H3, H4). We are given that a ball makes a ping if its diameter is less than or equal to the diameter of the hoop. Let the diameter of a ball be denoted as $D_B$ and the diameter of a hoop as $D_H$. A ping occurs when:

$D_B\le D_H$

Let's evaluate the number of pings on each hoop based on the given conditions.

Step 1: Pings on Hoop 1 (H1)
Condition 3 states that all balls, except B3, made pings on H1. This directly tells us exactly which balls successfully passed through H1: B1, B2, B4, B5, and B6. Therefore, the total number of pings on H1 is 5. It also tells us that B3 is larger than H1.

Step 2: Pings on Hoop 2 (H2)
Condition 4 states that none of the balls, except B2, made a ping on H2. This directly gives us the number of pings on H2. Only ball B2 passed through. Therefore, the total number of pings on H2 is 1. It also tells us that B2 is small enough to fit through H2, while all other balls are strictly larger than H2.

Step 3: Creating a size inequality chain
To determine the pings on hoops H3 and H4, we need to arrange the diameters of the balls and hoops in a chain of inequalities. Let's use Conditions 1, 2, and 4 to establish this.

From Condition 1: B1 and B6 ping on H4, but B5 does not. This gives us:

$D_{B5}>D_{H4}\ge D_{B1}$

From Condition 2: B4 pings on H3, but B1 does not. This gives us:

$D_{B1}>D_{H3}\ge D_{B4}$

From Condition 4: B2 pings on H2, but B4 does not. This gives us:

$D_{B4}>D_{H2}\ge D_{B2}$

Combining these three sets of inequalities, we form a master chain representing the relative sizes:

$D_{B5}>D_{H4}\ge D_{B1}>D_{H3}\ge D_{B4}>D_{H2}\ge D_{B2}$

Step 4: Pings on Hoop 4 (H4)
Now we determine exactly which balls pinged on H4 by looking at our chain. A ball will ping if it sits to the right of H4 in the inequality chain.
We can see that B1, B4, and B2 are all strictly smaller than or equal to H4. Additionally, Condition 1 explicitly states that B6 pings on H4.
What about the others? Condition 1 explicitly states B5 did not ping on H4. From Step 1, we also know B3 is the largest ball overall (larger even than B5 since it couldn't fit into the very large H1 hoop), so B3 definitely cannot fit into H4.
Therefore, exactly 4 balls ping on H4: B1, B2, B4, and B6. The total pings on H4 is 4.

Step 5: Pings on Hoop 3 (H3)
We do the same for H3. Looking to the right of H3 in our chain, we see that B4 and B2 are strictly smaller than or equal to H3, so both B4 and B2 ping on H3.
Looking to the left of H3, we see B1, B5, and B3 are all strictly larger than H3, meaning they definitely do not ping.
This accounts for 5 out of the 6 balls. We are missing B6. We know B6 is smaller than H4 and larger than H2, but we have no specific data linking its size relative to H3. B6 might be small enough to pass through H3, or it might be too large.
Therefore, the balls that ping on H3 are definitely B2 and B4, and possibly B6. The total pings on H3 is either 2 or 3.

Step 6: Total number of pings
We sum the number of pings across all four hoops:

• Hoop 1 (H1): 5 pings
• Hoop 2 (H2): 1 ping
• Hoop 3 (H3): 2 or 3 pings
• Hoop 4 (H4): 4 pings

Total = 5 + 1 + 4 + (2 or 3) = 12 or 13.

This is why the total number of pings from all tests undertaken is best described as 12 or 13.