A, suppose there is 1 sick dog, the owner of the sick dog will see that other dogs are not sick, so he knows that his dog is sick, so there will be a gunshot on the first night. Because there is no gunshot, it means that the number of sick dogs is greater than 1.

B suppose there are two sick dogs, and the owner of the sick dog will see 1 sick dog. Because the number of sick dogs exceeds 1 on the first day, the owner of the sick dog will know that his dog is sick, so there will be a gunshot the next day. The number of sick dogs was more than 2 due to the gunshot the next day.

Thus, if the gun goes off on the third day, there will be three sick dogs.

The second inference

1 If it is 1, the dog will die on the first day, because the dog owner did not see the sick dog, but the sick dog exists.

2 If it is 2, let the owners of sick dogs A and B, A see a sick dog and B also see a sick dog, but A sees that B's sick dog is not dead, so he knows that the number of dogs is not 1, and others have no sick dogs, so his dog must be a sick dog, so he shoots; And b and a thought the same, so they also shot.

So, at 2 o'clock, two dogs died on the first day.

3 If there are three dogs, let the dog owners A, B, C and A see two sick dogs on the first day. If A assumes that his dog is not a sick dog, and it is inferred that both dogs are not dead when he looks at it the next day, then the number of dogs is definitely not 2, and others are not sick dogs, then his dog must be a sick dog, so he shoots; And b and c thought the same as a, so they also made moves.

So, at 3 o'clock the next day, the three dogs died.

If there are four dogs, let the dog owners A, B, C, D and A see three sick dogs on the first day. If A assumes that his dog is not a sick dog and infers that three dogs are not dead on the third day, then the number of dogs is definitely not three, and others are not sick dogs, then his dog must be a sick dog, so he shoots; And b and c and d thought the same as a, so they also made moves.

So, at 4 o'clock, the last four dogs will die on the third day.

The rest is recursive, and n is derived from the year n- 1.

Answer: n is 4. On the fourth day, the dog died, but on the third day, so the answer is three.