Gem 13: The Twelve Coins Puzzle - Solutions
Solutions by Moshe Shweiger

	The Twelve Coins Puzzle
	We have twelve coins. They all weigh exactly the same, except for one which is fake. It is either heavier or lighter than the other eleven. Determine in three weighings of a balance scale which one of the coins is the fake one and whether it is heavier or lighter than the other eleven.

Short solution with consice wording and arrow notations:




If you are satified with this short type of notation,
that's great. But if you would like to see each
weighing with the details that lead to the actual
solution, even if it is a repetitive type of logic,
please proceed to the following longer solution
with different wording and illustrations:

	The Solution to the Twelve Coins Puzzle
	First weighing: Place four coins on each scale of the balance, and leave the remaining four coins aside. FIRST CASE - The scales are unbalanced as illustrated below: Mark the four coins on the light side with an upwards arrow meaning "Possibly Lighter", mark the four coins on the heavy side with a downwards arrow meaning "Possibly Heavier", and mark the four remaining coins with the word "GOOD" meaning that their weights are exactly the same since the fake coin must be on one of the scales. Second weighing (In the FIRST CASE): Place three coins on each scale as follows: Place two coins that are marked as "possibly lighter" and one coin marked as "possibly heavier" on one scale, and place one "GOOD" coin and one each of "possibly lighter" and "possibly heavier" coins on the other scale. This leaves one "possibly lighter" coin, two "possibly heavier" coins, and three "GOOD" coins aside. If the scales are unbalanced in this second weighing as illustrated below, perform a third weighing as follows: If the scale with the two "possibly lighter" coins is lighter as shown above, then the fake coin must either be one of the two "possibly lighter" coins from that lighter side or the one "possibly heavier" coin on the other heavier side. Now weigh one of those two "possibly lighter" coins against the other. If this third weighing is balanced, then the fake coin must be that "possibly heavier" coin, and it is heavier. (SOLVED) But if this third weighing is unbalanced, then the fake coin must be the one that is lighter, and it is lighter. (SOLVED) If the scale with two "possibly lighter" coins is heavier as shown below, then the fake coin must be either one of the "possibly heavier" coins from that heavier side or the one "possibly lighter" coin on the other lighter side. Now weigh that "possibly lighter" coin against a "GOOD" coin from the asides. If this third weighing is balanced, then the fake coin must be that "possibly heavier" coin not weighed, and it is heavier. (SOLVED) But if this third weighing is unbalanced, then the fake coin must be either the one that is lighter, and it is lighter (SOLVED), or the one that is heavier, and it is heavier. (SOLVED) If the scales are balanced in this second weighing as shown below, then the fake coin must either be one of the two "possibly heavier" coins from the asides or the one "possibly lighter" coin from the asides. Perform a third weighing as follows: Weigh the two "possibly heavier" coins from the asides against each other. If this third weighing is balanced, then the fake coin must be that "possibly lighter" coin from the asides, and it is lighter. (SOLVED) But if this third weighing is unbalanced, then the fake coin must be the one that is heavier, and it is heavier. (SOLVED) SECOND CASE - The scales are balanced in the first weighing as illustrated below: Mark the eight coins on the scales as "GOOD", and mark the four coins on the aside as "UNKNOWN" since they include the fake coin, and they must be checked in two more wieghings. Second weighing (In the SECOND CASE): Place two coins on each scale as follows: Place two unknown coins from the asides on one scale, and one "UNKNOWN" coin and one "GOOD" coin from the asides on the other scale. Now one "UNKNOWN" coin and seven "GOOD" coins remain in the asides. If the scales are unbalanced in this second weighing, perform a third weighing as follows: If the scale with the "GOOD" coin is lighter, then the fake coin must either be the "UNKNOWN" coin from that lighter side or one of the two "UNKNOWN" coins on the other heavier side. Now weigh one of those two "UNKNOWN" coins on the other heavier side against the other. If this third weighing is balanced, then the fake coin must be that "UNKNOWN" coin from that lighter side, and it is lighter. (SOLVED) But if this third weighing is unbalanced, then the fake coin must be the one that is heavier, and it is heavier. (SOLVED) If the scales are balanced in this second weighing, then the fake coin must be the "UNKNOWN" one from the scale with the "GOOD" coin, but we must determine whether it is lighter or heavier by the following third weighing: Weigh the fake coin against a "GOOD" coin. If the fake coin is lighter, it is lighter (SOLVED), but if the fake coin is heavier, it is heavier. (SOLVED) Moshe Shweiger

 

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