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More Coin Weighing: The McRiches Company


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Question

So the head of the McRiches Company in your town is looking for an advisor. The pay is great and the work light. This is just the sort of job you need! The position has been open though for quite some time. Why? Because to get hired, you have to pass a simple test. The details of this test are known by everyone, and are even printed out on the McRiches website:

 

Mr. McRiches will present the candidate with 14 US half dollars. One coin is fake, but it will weigh just slightly less or more than the others-- the candidate will not know which. One cannot tell the fake coin just by look or feel. The candidate will have a simple balance scale, but it only may be used 3 times. The successful candidate must be able to guarantee he/she will be able to tell Mr. McRiches which is the fake coin after only 3 weighings. If a candidate gets it wrong, Mr. McRiches will guarantee that candidate never gets a job anywhere ever again. Do not anger the man! He does not like to have his time wasted.

 

No one has even dared come into the office to try the test. The riddle seems impossible. But that was before this morning, You had a brainstorm last night, Today you waltzed to McRiches Headquarters and now you have a new job! Congratulations!

 

How did you do it?

 

 

Edited by Biotop
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  • Solution
Spoiler

Reminds me of the task of dividing 17 horses into groups of 1/2, 1/3 and 1/9: add a horse.

Spoiler

I would take a good coin (X) from my pocket and compare (X 1 2 3 4) with (5 6 7 8 9).

(1) If they balance, the fake is among (10 11 12 13 14).

Compare (X 1 2 all good) with say (10 11 12).

  • If they balance, the fake is 14 if (X) balances (13); otherwise it’s 13.
  • If (10 11 12) is lighter, one is a light fake. Comparing 10 with 11 says which.
  • If (10 11 12) is heavier, one is a heavy fake. Comparing 10 with 11 says which.

(2) If (X 1 2 3 4) is heavier, then 10-14 are good.

Compare (X 10 11 12 13 14 all good) with three coins from each side, say (2 3 4 and 5 6 7).

  • If they balance, the fake is among (1 8 9). Compare (1 8) with (X 10 both good).
              If (1 8) is lighter, then 8 is the light fake.
              If (1 8) is heavier, then 1 is the heavy fake.
  • If (2-7) is heavier, a heavy fake is in (2 3 4). Comparing 2 with 3 says which it is.
  • If (2-7) is lighter, a light fake is in (5 6 7). Comparing 5 with 6 says which it is. 

(3) if (X 1 2 3 4) is lighter, then 10-14 are good. Do (2), with the obvious changes.

 

 

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