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The odd coin (Posted on 2002-05-01) Difficulty: 3 of 5
In a pile, there are 11 coins: 10 coins of common weight and one coin of different weight (lighter or heavier). They all look similar.

Using only a balance beam for only three times, show how you can determine the 'odd' coin.

Open problem (i cannot solve this myself): how many more coins (with the same weight as the ten) can we add to that pile so that three weighing still suffices? My conjecture is zero, though my friend guessed that adding one is possible. The best bound we can agree upon is < 2.

  Submitted by theBal    
Rating: 3.1667 (6 votes)
Solution: (Hide)
Divide the 11 coins into three trios and one pair. Let the trios be A,B,C. Weigh A against B. Then weigh A against C. From this we can determine whether the odd coin lies in any of the trio or not.

Case 1: the odd coin does not lie in any trio. This happens when both weighings are balanced. Then the odd coin must be in the pair. Take any one coin from any trio (which are not odd), then weigh against any coin from the pair. If it shows balance, then the other coin from the pair is odd. If it is not balanced then the one on the balance (from the pair) is odd. Then we are done by three weighings.

Case 2: the odd one lies in the trio. If this is the case, then we can deduce which trio contains the odd coin, since, we have only three possibilities:
(i) A equal B, A unequal C => the odd coin is in C
(ii) A unequal B, A equal C => the odd coin is in B
(iii) A unequal B, A unequal C => the odd coin is in A

And also, we can easily check that we can determine whether the odd coin is lighter/heavier. Now take the trio where the odd coin is supposed to be. Then weigh any two of them: if it is equal then the third one is odd; if not then the lighter/heavier of them (depending on our observation earlier) is odd. We are done with three weighings.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): No SubjectLarryAllen2023-11-17 17:12:58
re(3): No SubjectAngus Hannaford2023-11-08 17:39:01
re(2): No SubjectAngus Hannaford2023-11-08 17:37:40
re(2): No SubjectAngus Hannaford2023-11-06 20:57:52
re: No SubjectAngus Hannaford2023-11-06 15:21:32
No SubjectLarryAllen2023-10-27 07:44:46
Puzzle ThoughtsK Sengupta2023-10-26 20:53:13
re: domyresume.netLarryAllen2023-10-26 17:00:59
domyresume.netLarryAllen2023-10-06 11:08:49
No SubjectCodyDunning2023-10-04 06:15:19
Magical tipsCodyDunning2023-08-31 07:23:23
magic trick toysCodyDunning2023-08-25 02:17:26
re: taxis welwynLarryAllen2023-08-17 05:05:04
re: taxis welwynLarryAllen2023-08-17 05:03:02
taxis welwynCodyDunning2023-05-29 03:04:48
re: Solution for 13 coins!!!Joel2006-11-12 23:21:03
re: Solution for 13 coins!!!Hugo2005-12-14 17:47:12
Solution for 13 coins!!!Morgan2005-12-14 14:31:03
re: 12 coins conclusive answer (now with 13!)Morgan2005-12-14 14:29:42
Some Thoughtsre: you can do up to 27 with 3 times weighingron2004-12-31 16:53:07
you can do up to 27 with 3 times weighingAli2004-01-05 01:57:49
re(2): this is easy..Sanjay2003-06-06 14:16:37
re: this is easy..me2003-06-05 23:05:41
this is easy..me2003-06-05 22:59:02
SolutionAnother 12 coin solutionSanjay2003-05-11 03:28:22
Questionre a better open problemJonathan2003-03-17 15:07:20
insaneJonathan2003-03-17 15:00:19
maybe!!Jay2003-03-16 11:35:36
re: a better open problemnikki2003-03-06 09:32:27
12 coins conclusive answerJohn Smith2003-02-10 10:53:16
re: answerTomM2002-10-23 10:27:31
answerI2002-10-23 09:16:37
Question12 coinsAeternus2002-10-11 04:38:29
non-adaptiveCheradenine2002-06-13 00:14:48
12 coin colution (continued)Half-Mad2002-05-16 02:20:02
12 coin solutionHalf-Mad2002-05-16 02:19:28
12 (almost)levik2002-05-03 07:21:36
a better open problemtheBal2002-05-02 10:00:06
to maxtheBal2002-05-02 09:26:52
Max number of coinsDaniel2002-05-02 08:42:22
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