## Tuesday, January 19, 2010

### The Scale, Parts 1 and 2

Part 1: You have 8 identical-looking gold coins; 7 are equal weight, 1 is counterfeit and slightly heavier.  Using an old-fashioned balance scale only twice, find the counterfeit coin.

Part 2: Consider 8 gold coins again.  However, this time it is unknown whether the counterfeit coin is heavier or lighter.  You can use the scale 3 times.  Find the counterfeit coin.

Solutions:
• Part 1:
• Weigh-in 1: Put 3 coins on one side, 3 coins on the other; leave 2 coins aside.  The counterfeit coin is on the heavier side.  If both sides are equal, the counterfeit coin is one of the 2 that was left aside.
• Weigh-in 2: If the counterfeit coin was in a group of 3, take 2 of those 3 and put one on each side.  The heavier side bears the counterfeit.  If the sides are equal, the counterfeit coin was the one left out.  If the counterfeit coin was in a group of 2, just weigh the two against each other and the heavier one is counterfeit.
• Part 2:
• Weigh-in 1: Same as above.  If the sides are unequal, however, the counterfeit could be on either side.
• Weigh-in 2: If the sides are unequal, take one coin from the heavy side, one coin from the light side and put it on the left tray.  Take another coin from the heavy side and another coin from the light side and put it on the right tray.  So you have 1 heavy, 1 light on each side.  Now depending on which side is heavier, you can narrow down to 2 suspects.
• Weigh-in 3: Now that you only have 2 suspects, compare 1 of them to a known authentic coin.  If it is equal in weight to the normal coin, then the other is counterfeit.  If unequal, then it is the counterfeit.
• A sample grouping is demonstrated below.  There are several possible choices depending on intermediate outcomes, but in this case, let the left side always be heavier.  X indicates unknown, H means the coin is potentially too heavy, L means the coin may be too light, and O means the coin is known to be authentic.
• 1: XXX | XXX       XX
• result: HHH | LLL        OO
• 2: HL | HL           OO HL
• result: HO | OL      OO OO
• 3: H | O      L OOOOO
• result: H | O   OOOOOO