12th ball problem

A difficult problem occurred when I sent a text message at night, as described below:

There is a bad ball in the twelve balls. Its weight is different from the rest, and its appearance is no different,
You only have a balance without weight. It is called three times. Can you find it?

[Rough solution]
A rough look is probably divided into three groups A (1, 2, 3, 4) B (5, 6, 7, 8) C (9, 10, 11, 12) Better,
Four balls in each group. (because there is no hope for six or three balls in each group, I have a hunch)
The first a vs B results in two types: A = B (equal weight)! = B (not equal)
1. If a = B, the bad ball is in C. Use two good balls vs (9, 10). If they are equal, they are in (11, 12.
We can use a good ball vs (11) to get the answer. If it doesn't wait, we can obviously find out.
2. If! = B set A> B (A over B), then the bad ball in a (,) or B (,) is divided into four groups. A (1, 2) B (3, 4) C (5, 6) D (7, 8)
Take a (1, 2) vs C (5, 6). If they are equal, they are in B and D. They are not equal in A and C.
(1) equal. In B (3, 4) and D (7, 8), replace with a normal ball. 3. Adjust 4 and 7. That is, ())
If equal, 3 is the request. If>, 8 is the request. If <, G is the request.
(2) not equal. Likewise.

The above solution is incorrect.
In this sentence, if the value is equal, 3 is the request. If> 8 is the request. If <G is the request. In fact, G and D can be the request.

Correct answer:

The first step is the same.
In the second step, you only need to select (, 5) and (, 6). If they are equal, they are in (). Obviously.
If the preceding (1, 2, 3, 4)> (5, 6, 7, 8)
So
If (, 5)> (, 6), the bad ball can only be in, 6 (carefully understood). and 1 or 2 may be heavy or 6 is light.
When (1) vs (2) is equal, it is 6.> 1, <2.
If (1, 2, 5) <(3, 4, 6), the bad ball may only be in 3, 4, 5, and 3 or 4 may be a heavy ball or 5 is a light ball.
Similarly, when (3) vs (4) is equal, it is 5.> 3, <4.

Solution completed.

Features of this question: give full play to the effectiveness of the information obtained by each weighing, such as bias, and fully grasp the relationship between weighing and unweighing.
Use the subtle links between them to solve problems.