Divide those 12 balls into 3 groups, each having 4 balls. Name them as group A, B, and C. Weigh 1: Let's put A and B on each side of the balance. We will have the following three results: 1a: Weight(A) > Weight(B) 1b: Weight(A) = Weight(B) 1c: Weight(A) < Weight(B) Since 1a and 1c are actually equivalent, we only need to study case 1a and 1b. (1a and 1c can be switched by changing the name of two groups.)Case 1a: Let's give names to each balls in group A and B. They are A1, A2, A3, A4, B1, B2, B3, and B4. We have Weight(A1, A2, A3, A4) > Weight(B1, B2, B3, B4). We also know that balls in Group C are all normal balls. Let's call them C1, C2, C3 and C4. Weigh 2(1): This step is critical. Put ball A1, A2, and A3 aside. Put ball A4, B1, B2, and B3 on one side of the balance, and B4, C1, C2, and C3 on the other side of the balance. We will have the following three results: 2a: Weight(A4, B1, B2, B3) > Weight(B4, C1, C2, C3) 2b: Weight(A4, B1, B2, B3) = Weight(B4, C1, C2, C3) 2c: Weight(A4, B1, B2, B3) < Weight(B4, C1, C2, C3) Case 2a: In this case, we know there are two possible results. Namely, either A4 is fake and heavier, or B4 is fake and lighter. Weigh 3(1): Let's weigh A4 and any other normal balls this time. We will have the following two results: A4 is heavier or A4 is equal to an normal ball, which tells us B4 is lighter. Case 2b: In this case, we know the fake ball is in (A1, A2, A3) and it is heavier than normal ones. We can easily find it using one weigh. Weigh 3(2): Put A1 and A2 on the balance. If balance, A3 is the fake ball. If not balance, the heavier one is the fake ball. Case 2c: In this case, we know that the fake ball is in (B1, B2, B3) and it is lighter than normal ones. Similarly, we can find it using one weigh. Weigh 3(3): Put B1 and B2 on the balance. If balance, B3 is the fake ball. If not balance, the lighter one is the fake ball.By now, we finished discussion of Case 1a.Case 1b: We know the fake ball is in group C, but we don't know whether it is heavier or lighter. We name the balls in the same way as we did just now. Weigh 2(2): Put A1, A2 and A3 on one side of the balance. Put C1, C2, C3 on the other side of the balance. We will have the following three results: Case 2x: Weight(A1, A2, A3) > Weight(C1, C2, C3) Case 2y: Weight(A1, A2, A3) = Weight(C1, C2, C3) Case 2z: Weight(A1, A2, A3) < Weight(C1, C2, C3) Case 2x: In this case, we know the fake ball is in (C1, C2, C3) and it is lighter. We can find it using one more weigh. Case 2y: In this case, we know the fake ball is C3. Another weigh will tell us whether it is heavier or lighter than normal ones. Case 2z: In this case, we know the fake ball is in (C1, C2, C3) and it is heavier. We can find it using one more weigh.So we can finish the discussion of Case 1b.By now, we can conclude that we can find the fake ball in three weighs.//
第二次:分二份,第份二个------->去掉二个
第三次:--------------------->去掉一个
得答案:
Weigh 1: Let's put A and B on each side of the balance. We will have the following three results:
1a: Weight(A) > Weight(B)
1b: Weight(A) = Weight(B)
1c: Weight(A) < Weight(B) Since 1a and 1c are actually equivalent, we only need to study case 1a and 1b. (1a and 1c can be switched by changing the name of two groups.)Case 1a:
Let's give names to each balls in group A and B. They are A1, A2, A3, A4, B1, B2, B3, and B4.
We have Weight(A1, A2, A3, A4) > Weight(B1, B2, B3, B4).
We also know that balls in Group C are all normal balls. Let's call them C1, C2, C3 and C4. Weigh 2(1): This step is critical. Put ball A1, A2, and A3 aside. Put ball A4, B1, B2, and B3 on one side of the balance, and B4, C1, C2, and C3 on the other side of the balance. We will have the following three results:
2a: Weight(A4, B1, B2, B3) > Weight(B4, C1, C2, C3)
2b: Weight(A4, B1, B2, B3) = Weight(B4, C1, C2, C3)
2c: Weight(A4, B1, B2, B3) < Weight(B4, C1, C2, C3) Case 2a: In this case, we know there are two possible results. Namely, either A4 is fake and heavier, or B4 is fake and lighter.
Weigh 3(1): Let's weigh A4 and any other normal balls this time. We will have the following two results:
A4 is heavier or A4 is equal to an normal ball, which tells us B4 is lighter. Case 2b: In this case, we know the fake ball is in (A1, A2, A3) and it is heavier than normal ones. We can easily find it using one weigh.
Weigh 3(2): Put A1 and A2 on the balance. If balance, A3 is the fake ball. If not balance, the heavier one is the fake ball. Case 2c: In this case, we know that the fake ball is in (B1, B2, B3) and it is lighter than normal ones. Similarly, we can find it using one weigh.
Weigh 3(3): Put B1 and B2 on the balance. If balance, B3 is the fake ball. If not balance, the lighter one is the fake ball.By now, we finished discussion of Case 1a.Case 1b: We know the fake ball is in group C, but we don't know whether it is heavier or lighter. We name the balls in the same way as we did just now.
Weigh 2(2): Put A1, A2 and A3 on one side of the balance. Put C1, C2, C3 on the other side of the balance. We will have the following three results:
Case 2x: Weight(A1, A2, A3) > Weight(C1, C2, C3)
Case 2y: Weight(A1, A2, A3) = Weight(C1, C2, C3)
Case 2z: Weight(A1, A2, A3) < Weight(C1, C2, C3) Case 2x: In this case, we know the fake ball is in (C1, C2, C3) and it is lighter. We can find it using one more weigh.
Case 2y: In this case, we know the fake ball is C3. Another weigh will tell us whether it is heavier or lighter than normal ones.
Case 2z: In this case, we know the fake ball is in (C1, C2, C3) and it is heavier. We can find it using one more weigh.So we can finish the discussion of Case 1b.By now, we can conclude that we can find the fake ball in three weighs.//
拿出其中两份放到天平两侧称（第一次）
情况一：天平是平衡的。
那么那八个拿上去称的小球都是正常的，特殊的在四个里面。
把剩下四个小球拿出三个放到一边，另一边放三个正常的小球（第二次）
如天平平衡，特殊的是剩下那个。
如果不平衡，在天平上面的那三个里。而且知道是重了还是轻了。
剩下三个中拿两个来称，因为已经知道重轻，所以就可以知道特殊的了。（第三次）
情况二：天平倾斜。
特殊的小球在天平的那八个里面。
把重的一侧四个球记为A1A2A3A4，轻的记为B1B2B3B4。
剩下的确定为四个正常的记为C。
把A1B2B3B4放到一边，B1和三个正常的C小球放一边。（第二次）
情况一：天平平衡了。
特殊小球在A2A3A4里面，而且知道特殊小球比较重。
把A2A3称一下，就知道三个里面哪个是特殊的了。（第三次）
情况二：天平依然是A1的那边比较重。
特殊的小球在A1和B1之间。
随便拿一个和正常的称，就知道哪个特殊了。（第三次）
情况三：天平反过来，B1那边比较重了。
特殊小球在B2B3B4中间，而且知道特殊小球比较轻。
把B2B3称一下，就知道哪个是特殊的了。（第三次）
情景1
1 =2组 （第一次） 3 >4组或 3<4 （第二次）比较第一次和第二次 知道是哪一组而且是重还是轻
情景2
1>2 或1<2 （第一次） 3=4（第二次） 比较第一次和第二次 知道是哪一组而且是重还是轻
然后再把有问题的那组 分为1球 2球 3球（第三次）
1球=2 则为3
1=!2则 根据 情景1或情景2 可以判断是哪个是轻还是重
第二次:分二份,第份二个------->去掉二个
第三次:--------------------->去掉一个
这也太简单了，不过在现场回答可能会一时想不到。
步骤：
1 . 12球分成两组（6:6），挑出轻的一组。
2. 将轻的一组再分成两组（3:3），挑出轻的一组
3. 将最后一组随便挑出两个称一下不就知道了吗。
OVER !!!!!!!