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

The correct answer is 6.

To determine the total number of pings made by balls B1, B2, and B3, we can analyze the given rules to figure out the relative sizes of the balls and hoops. Recall that a ball makes a "ping" if its diameter is less than or equal to the hoop's diameter.

Let us first determine the number of pings made by B1:

From Rule 3, all balls except B3 made a ping on H1, which means B1 made a ping on H1.
From Rule 4, only B2 made a ping on H2, which means B1 did not make a ping on H2.
From Rule 2, we are explicitly told that B1 did not make a ping on H3.
From Rule 1, we are explicitly told that B1 made a ping on H4.

Therefore, B1 made a total of 2 pings (on H1 and H4).

Next, let us determine the number of pings made by B2:

From Rule 3, B2 made a ping on H1.
From Rule 4, B2 made a ping on H2.
Rule 4 also tells us that no other ball made a ping on H2, which implies that B2 is the smallest ball and H2 is the smallest hoop. Let's prove that B2 also fits in H3 and H4. From Rule 2, B4 pings on H3, meaning H3 is large enough to fit B4. Since B4 is too large for H2 (from Rule 4), H3 must be larger than H2. Because B2 can pass through H2, it can definitely pass through the larger hoop H3. Similarly, Rule 1 states B1 pings on H4, meaning H4 is large enough to fit B1. Since B1 is too large for H2, H4 must also be larger than H2. Because B2 fits perfectly fine in H2, it also passes easily through the larger hoop H4.

Therefore, B2 made a total of 4 pings (on H1, H2, H3, and H4).

Now, let us determine the number of pings made by B3:

From Rule 3, B3 explicitly did not make a ping on H1.
From Rule 4, B3 did not make a ping on H2.
Rule 3 also tells us that all other balls pinged on H1, which implies B3 is the largest ball and H1 is the largest hoop. Let's prove that B3 cannot fit in H3 or H4 either. From Rule 2, B1 cannot fit through H3, meaning H3 is smaller than B1. Since B1 fits through H1 (from Rule 3), H3 must be strictly smaller than H1. Because B3 is too large to fit through H1, it is definitely too large to fit through the smaller hoop H3. Similarly, Rule 1 states B5 cannot fit through H4, meaning H4 is smaller than B5. Since B5 fits through H1 (from Rule 3), H4 must be strictly smaller than H1. Because B3 is too large for H1, it is also too large for the smaller hoop H4.

Therefore, B3 made a total of 0 pings.

Finally, we sum the successful pings for the three specified balls (B1, B2, and B3):

$2+4+0=6$

The total number of pings made by B1, B2, and B3 is 6.