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One of Sixteen (Posted on 2015-12-29) Difficulty: 3 of 5
I have two sets of 8 coins. In one set the coins weigh 30g each, in the other set the coins weigh 31g each.
Unfortunately they got mixed together in one big pile of 16 coins. I want to identify one coin. It can be from either set.

(Easy) Using a balance scale, identify a coin in four weighings.

(Hard) Identify a coin in just three weighings.

  Submitted by Brian Smith    
Rating: 4.0000 (3 votes)
Solution: (Hide)
(Easy) Four weighing solution:
Weigh 8 vs 8. If unequal keep the heavier side, if equal keep either side.
Split that side into halves and weigh 4 vs 4. Again, if unequal keep the heavier side, if equal keep either side.
Repeat for 2 vs 2 and 1 vs 1. The single coin picked at the end will be from the heavy set.

(Hard) Three weighing solution:
Weigh 7 vs 7 with 2 coins left off.

Case 1: 7 vs 7 is unequal.
- The heavy side has at least 4 heavier coins, Weigh 3 vs 3 from that side. If unequal the heavier side has at least two 31 g coins, take that side. If equal both sides have at least two heavy coins, take either side.
- Make the last weighing 1 vs 1 from the group of three chosen. Either the weighing is unequal which gives a heavy and light coin or the weighing is equal which means both are heavy.

Case 2: 7 vs 7 is equal.
- Both sides contain 3 light and 4 heavy or both contain 4 light and 3 heavy, the two coins left off must be of equal weight. Choose one side and weigh 3 vs 3 with the last coin left off.

Subcase 2-1: 3 vs 3 is unequal.
- The heavy side must have at least two heavy coins. Make the third weighing 1 vs 1 similar to Case 1.

Subcase 2-2: 3 vs 3 is equal.
- Both sides contain 1 light and 2 heavy or both contain 2 light and 1 heavy. Furthermore the coin left off the second weighing must weigh the same as the two coins left off in the first weighing. Make a set of three from these left off coins and compare that set to one of the sets of three from the second weighing. That weighing will determine what weight the three previously left off coins weigh.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(4): Short solution.broll2016-01-12 22:14:47
re(3): Short solution.Jer2016-01-12 10:20:32
Some Thoughtsre(2): Short solution.broll2016-01-11 23:08:08
re: Short solution.Jer2016-01-11 12:30:33
re(2): What's the problem? (and a spoiler)Steve Herman2016-01-01 10:34:53
re: What's the problem? (and a spoiler)Brian Smith2015-12-31 13:06:07
QuestionWhat's the problem? (and a spoiler)Steve Herman2015-12-31 12:16:44
Some Thoughtsre: Short solution.Ady TZIDON2015-12-31 10:49:14
Short solution.broll2015-12-30 22:09:31
re: Solution (hard)Ady TZIDON2015-12-30 17:01:14
SolutionSolution (hard)Paul2015-12-30 12:08:39
Some ThoughtsEasy does itSteve Herman2015-12-29 13:45:30
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