回複:用天平稱乒乓球

來源: 異言 2010-05-09 16:38:15 [] [博客] [舊帖] [給我悄悄話] 本文已被閱讀: 次 (4011 bytes)
那麽多自稱容易和已解出的,可我好像沒有看出正確且完全的12球三次解。哪個是?請指出?更不要說24球三次解了。

本人的12球三次解:(懇請驗證者指出疏漏):
將2球分為三組,A,B,C,每組四球, 編號為:
A1,A2,A3,A4,B1,B2,B3,B4,C1,C2,C3,C4.

第一稱:
A1,A2,A3,A4 vs. B1,B2,B3,B4

三種可能的結果:
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. (請記住這些結論,我們後麵要多次用到它們)

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.)

以結果不同分別進行第二稱:
-----------------------------------------------------
第二稱 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 第三稱: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 第三稱:C1 vs C2. We may get final result in the symetric way as last paragraph.

---------------------------------------------------
第二稱 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).
(第三稱: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 第二稱 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 第三稱: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 第二稱 A1,A2,B1 vs. C1,A3,B2)
What we may get is: bad ball could either B1 (lighter), or A3 (heavier).
Then going to 第三稱 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.

---------------------------------------------------
第二稱 for result 3 of first weighing:
sysmetrically similar to result 2 of first weighing.
...get the details yourselves by following the 第二稱 for result 2.

---------------------------------------------------
我的解答對聰明人可能顯得有些羅嗦。歡迎指正。
有興趣的朋友可以舉一反三試試24球三次解。




所有跟帖: 

回複:回複:你的方法真的很笨 -螞蟻王子- 給 螞蟻王子 發送悄悄話 (214 bytes) () 05/09/2010 postreply 19:26:56

回複:回複:回複:你的方法真的很笨 -異言- 給 異言 發送悄悄話 異言 的博客首頁 (36 bytes) () 05/09/2010 postreply 21:15:49

回複:回複:回複:回複:你的方法真的很笨 -螞蟻王子- 給 螞蟻王子 發送悄悄話 (153 bytes) () 05/10/2010 postreply 16:42:19

回複:回複:回複:你的方法真的很笨 -vixie- 給 vixie 發送悄悄話 (165 bytes) () 05/21/2010 postreply 13:07:29

加跟帖:

當前帖子已經過期歸檔,不能加跟帖!