The traditional type of scale allowed for the weighings While living in Manhattan I met a wonderfully brilliant scientist named Colleen Noviello. Besides being my own personal tour guide to the city and even up into New York, she was always interested in challenging me with puzzles and games of strategy (chess for example).
One afternoon she presented me with what she claimed to be one of the most difficult logic puzzles she had ever attempted. Evidently her father had given her this at one point. I honestly can't remember if she ever managed to solve it or not, but I spent the next couple of days playing with it and much to my joy, eventually solving it.
The puzzle and my solution are below.
Puzzle: There are 12 balls. One of them is either heavier or lighter then the other 11. Assume that you have access to a traditional balance scale (one side is weighed against the other). Using only three weighing, determine which ball is off weight AND if it is heavier or lighter than the rest of the balls.
Solution for a 4x4x4 grouping:
Start by numbering the balls 1 2 3 4 5 6 7 8 9 10 11 12. Because they can be rearranged in any order without losing our principle, we can defer to the standard mathematical concept of WLOG (without loss of generality).
Step One
First Possibility: 1 2 3 4 > 5 6 7 8 1,2,3,4 are greater or 5,6,7,8 are lighter (9,10,11,12 are equal)
Second Possibility: 1 2 3 4 < 5 6 7 8 1,2,3,4 are lighter or 5,6,7,8 are heavier (9,10,11,12 are equal)
Third Possibility: 1 2 3 4 = 5 6 7 8 9,10,11,12 are either heavier or lighter (1,2,3,4,5,6,7,8 are equal)
Step Two of First Possibility
First Possibility: 1 2 5 11 > 3 4 6 12 1,2 are heavier or 6 is lighter
Second Possibility: 1 2 5 11 < 3 4 6 12 5 is lighter or 3,4 are greater
Third Possibility: 1 2 5 11 = 3 4 6 12 7,8 are lighter
Step Two of Second Possibility
First Possibility: 5 6 1 11 > 7 8 2 12 5,6 are greater or 2 is lighter
Second Possibility: 5 6 1 11 < 7 8 2 12 1 is lighter or 7,8 are greater
Third Possibility: 5 6 1 11 = 7 8 2 12 3,4 are lighter
Step Two of Third Possibility
First Possibility: 9 10 1 > 11 2 3 9,10 are greater or 11 is lighter
Second Possibility: 9 10 1 < 11 2 3 9,10 are lighter or 11 is greater
Third Possibility: 9 10 1 = 11 2 3 12 is either lighter or heavier
Step Three of First Possibility of First Possibility
First Possibility: 1 6 > 11 12 1 is the heavier ball
Second Possibility: 1 6 < 11 12 6 is the lighter ball
Third Possibility: 1 6 = 11 12 2 is the heavier ball
Step Three of Second Possibility of First Possibility
First Possibility: 3 5 > 11 12 3 is the heavier ball
Second Possibility: 3 5 < 11 12 5 is the lighter ball
Third Possibility: 3 5 = 11 12 4 is the heavier ball
Step Three of Third Possibility of First Possibility
First Possibility: 7 < 12 7 is the lighter ball
Second Possibility: 7 = 12 8 is the lighter ball
Step Three of First Possibility of Second Possibility
First Possibility: 5 2 > 11 12 5 is the heavier ball
Second Possibility: 5 2 < 11 12 2 is the lighter ball
Third Possibility: 5 2 = 11 12 6 is the heavier ball
Step Three of Second Possibility of Second Possibility
First Possibility: 1 7 > 11 12 7 is the heavier ball
Second Possibility: 1 7 < 11 12 1 is the lighter ball
Third Possibility: 1 7 = 11 12 8 is the heavier ball
Step Three of Third Possibility of Second Possibility
First Possibility: 3 < 12 3 is the lighter ball
Second Possibility: 3 = 12 4 is the lighter ball
Step Three of First Possibility of Third Possibility
First Possibility: 9 11 > 1 2 9 is the heavier ball
Second Possibility: 9 11 < 1 2 11 is the lighter ball
Third Possibility: 9 11 = 1 2 10 is the heavier ball
Step Three of Second Possibility of Third Possibility
First Possibility: 9 11 > 1 2 11 is the heavier ball
Second Possibility: 9 11 < 1 2 9 is the lighter ball
Third Possibility: 9 11 = 1 2 10 is the lighter ball
Step Three of Third Possibility of Third Possibility
First Possibility: 12 > 1 12 is the heavier ball
Second Possibility: 12 < 1 12 is the lighter ball
Solution started on Thursday July 22, 1999 @ approximately 12:00 AM EST
Solution completed on Sunday July 25, 1999 @ 7:44PM EST
Time of solution was approximately 3 days 19 hours and 44 minutes


