Tuesday, March 9, 2010

The Scale, Part 3

Told to me by Tommy Dickie:

This time you have 12 identical-looking gold coins.  11 are equal weight, 1 is counterfeit and you don't know whether it's heavier or lighter than the others.  Using an old-fashioned balance scale only three (3) times, find the counterfeit coin and determine whether it is lighter or heavier than the others.  

I forget how to do this one!  Can someone help by submitting in the comments section?  Don't worry, there's definitely a solution and I'll get to it when I have time.  

I figured it a solution, although I don't think it's the only one.  I wrote the solution below, but it probably doesn't make any sense.  It would be easiest to call me or maybe video chat if you want a better understanding of how to solve it.  

I will try to represent graphically: let 'X' represent each unknown weight.  and the weights in brackets [XX|XX] will represent those on the left and right sides of the scale, respectively.  There are a variety of possible scenarios in this solution, but all involve no more than 3 weigh-ins.  

We begin by placing four weights on each side of the scale and leaving 4 off the scale, represented as follows:


Either the scale will be balanced, or unbalanced.  I treat both possibilities in outline format below:

  1. The scale is unbalanced.  Rename the suspect weights on the heavier side 'H' and those on the lighter side 'L'.  Known authentic weights will be renamed 'O'.  The scale now looks like this: [HHHH|LLLL] OOOO.  Next we will place 2 H's and 2 L's on the left side and 1 H, 1 L, and 2 Os on the right, as follows: [HHLL|HLOO] HLOO.  From here, there are three possibilities:
    1. The left side is heavier, in which case either one of the H's on the left is heavy, or the L on the right is light.  So weigh as follows: [H|H] LOOOOOOOOO. Either the left H is heavy, the right H is heavy, or they are both authentic (the scale is balanced), in which case the L is light. 
    2. If the left side is lighter, we weigh as follows: [L|L] HOOOOOOOOO.
  2.  If the scale is balanced, then all of the weights that were on the scale can be confirmed authentic, and the situation looks like this: [OOOO|OOOO] XXXX.  We will weigh as follows: [XXX|OOO] XOOOOO.  We will have 3 possibilities:
    1. The left side is heavier [HHH|OOO] OOOOOO.  Then we weigh as follows:
      [H|H] HOOOOOOOOO.  
    2. The left side is lighter [LLL|OOO] OOOOOO.  Then we weight as follows:
    3. The scale is even [OOO|OOO] XOOOOO.  Then we simply weigh the unknown against a known authentic O: [X|O] OOOOOOOOOO. 

1 comment:

  1. Although fairly obvious, you should include the 3rd possibility on your 2nd weighing: that the [HHLL|HLOO]HLOO balances.
    3. If it balances, the H or L left off the scale is counterfeit. Then weigh [H|O]LOOOOOOOOO. If left goes down, that H is it and if it balances the L is it.