回复：用天平称乒乓球

那么多自称容易和已解出的，可我好像没有看出正确且完全的12球三次解。哪个是？请指出？更不要说24球三次解了。

本人的12球三次解：（恳请验证者指出疏漏）：
将2球分为三组，A,B,C，每组四球， 编号为：
A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3,C4.

第一称：
A1,A2,A3,A4 vs. B1,B2,B3,B4

三种可能的结果：
result 1: group A = group B (the bad ball is in group C, and it could be either heavier or ligher one).

result 2: group A > group B (the bad ball is in group A or group B; If in group A, it is a heavier one, if in group B, it is a ligher one. （请记住这些结论，我们后面要多次用到它们)

result 3: group A < group B (the bad ball is in group A or group B; If in group A, it is a lighter one, if in group B, it is a heavier one.)

以结果不同分别进行第二称：
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第二称 for result 1 (Group A = Group B):
A1,C1,C2 vs. A2,A3,C3 (A4, and all B balls are good ones and out. Also, C4 is out but it could be the ball.)

result 1: both sides are same. then the bad ball must be the non-measured C4 ball. weigh C4 with any good ball and know whether it is lighter or heavier, resolved.

result 2: left is heavier than right. then what we can get is: C4 is a good ball; If the bad ball is ligher it must be C3; If the bad ball is heavier, it must be either C1 or C2. With this in mind going to 第三称：C1 vs C2. If same, then the bad ball is C3 which is lighter. If C1 is lighter than C2, then bad ball is C2 (heavier); If C1 is heavier than C2, then the bad ball is C1 (heavier) because we knew in previous step that C1 or C2 is heavier than normal ones. Resolved by now.

result 3: left is ligher than right. then what we can get is: If the bad ball is ligher it must be C1 or C2; if the bad ball is heavier, it must be C3. go to 第三称：C1 vs C2. We may get final result in the symetric way as last paragraph.

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第二称 for result 2 of first weighing (group A is heavier than group B):

A1,A2,B1 vs. C1,A3,B2 (with A4, B3, and B4 out)

3 results again:
result 1: same
Then, the bad ball must be among A4, or B3 or B4 (they were out in second weighing).
(第三称：B3 vs. B4
3 results again:
result 1: same weight. Then A4 is bad ball and lighter.
result 2: B3 is heavier than B4. Then B4 must be the bad and lighter (we knew from first weighing that if the bad ball is a B ball it must be a lighter one .)
result 3: B3 is lighter than B4. Then B3 must be the bad ball and lighter.) by now all cases with bad ball among A4, or B3 or B4 are resolved.

result 2: left is heavier than right (after 第二称 A1,A2,B1 vs. C1,A3,B2)
Then we may know bad balls are among A1 (heavier), or A2 (heavier), or B2 (lighter). going to 第三称：A1 vs. A2. If same, the bad ball is B2 lighter. If A1 is heavier than A2, then the bad ball is A1. If A1 is lighter than A2, then the bad ball is A2, becasue we knew from first weighing that if the bad ball is in group A, it must be a heavier one. By now all cases in result 2 of second weighing are resolved.

result 3: left is lighter than right (after 第二称 A1,A2,B1 vs. C1,A3,B2)
What we may get is: bad ball could either B1 (lighter), or A3 (heavier).
Then going to 第三称 A3 vs. C1.
If A3 is heavier, then it must be the bad ball.
If A3 is same as C1,B1 is the bad ball and lighter.
A3 being lighter than C1 is impossible since we knew bad A must be a heavier one in first weighing.
So far all cases in result 2 of first weighing are resolved.

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第二称 for result 3 of first weighing:
sysmetrically similar to result 2 of first weighing.
...get the details yourselves by following the 第二称 for result 2.

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我的解答对聪明人可能显得有些罗嗦。欢迎指正。
有兴趣的朋友可以举一反三试试24球三次解。

所有跟帖： 

• 回复：回复：你的方法真的很笨 -蚂蚁王子- ♀ (214 bytes) () 05/09/2010 postreply 19:26:56

• 回复：回复：回复：你的方法真的很笨 -异言- ♂ (36 bytes) () 05/09/2010 postreply 21:15:49

• 回复：回复：回复：回复：你的方法真的很笨 -蚂蚁王子- ♀ (153 bytes) () 05/10/2010 postreply 16:42:19

• 回复：回复：回复：你的方法真的很笨 -vixie- ♀ (165 bytes) () 05/21/2010 postreply 13:07:29