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Solution to Puzzle 113.  COUNTERFEIT COIN

You have 25 gold coins.  One on these coins is a counterfeit and weighs lighter, while the rest of the coins have the same weight.  If you have a balance, how would you determine which coin is the lighter one with 3 weighings. 


Divide the coins into 3 group consisting of 9 coins, 9 coins, and 7 coins.  On the first weighing, weigh the 2 groups of nine coins.  If they have the same weight, then the counterfeit is in the group with 7 coins.  If one of the group of 9's weighs less, then the counterfeit is on that group.

On the second weighing, divide the group where you know the counterfeit coin is into 3 subgroups, where the first two subgroups would have 3 coins each.  Weigh these first two subgroups.  If they have the same weight then the counterfeit is on the third subgroup.  If one of them weighs less, the counterfeit is on the subgroup that weighs less.

Since the subgroup that contains the counterfeit has at most 3 coins.  Weighing two of these coins will allow you to know which coin is the counterfeit.


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Bits and Beyond...


Pi is the ratio of the circumference of a circle to its diameter.  The approximation of pi has began as early as the ancient times.  One of the well-known one was by the great Greek mathematician Archimedes, who started approximating Pi by inscribing a hexagon into a circle.  With the hexagon, pi was calculated to have a value of 3.  Doubling the number of sides and inscribing a dodecagon in the same circle, his new value became closer to pi.  He continued until he used a 96-gon, and found a better approximation of 3.1419.  A similar method was used by the Chinese mathematician Lui Hui.  He used a 3,072-gon and acquired a value of 3.1416, a very good approximation approximation.