Balls and Scales riddle

[herbbread]

Hi all, I'm new to these forums, but I absolutely love riddles. I don't know if this one has been posted, but I love it and it took me ages to figure out. I'm sure someone can figure it out much more quickly.

There are 12 balls. All of them weigh exactly the same except for one. That odd ball out is either heavier or lighter than the rest. You have a balance scale (which can tell you which side has the heavier items) and are permitted to make only 3 weighings (i.e. make comparisons between any two sets of balls at most 3 times). How do you weigh the balls so you that can figure out (with 100% certainty) which ball is different and whether it is heavier or lighter than the rest?

[Schrödinger's hat]

I know I've solved this in the past, but there's one case the following approach does not account for.

 

 

1	Split the balls into groups of four, weigh them against each other

2If they weigh the same, grab any three and weigh them in the left pan against three of the remaning four in the rightIf they are different, grab any three from the side that was heavier, put them in the left pan with one ball from the side that was lighter. Weigh them against the four un-weighed balls

3If these are the same: you know it is the one you have not weighed yet. Weigh it against any ball to know if it's heavier. Else:

If it goes up, it's the single ball from the light side.

If it goes down, it's one of the three from the heavy side and it's heavy. Weigh two of the three, it's the heavy one if they are different, the remaining one if they are the same.

If weigh two shows equal, it's one of your remaining four (one from the heavy set, three from the light)

If they are different, it's one of the three in the right, if they go up it's light and vice versa. Grab two of these three and weigh them against one another. Pick the one that goes in the same direction, this is your ball and you know if it's heavier or lighter from the prev. step.

If they are the same it's the remaining ball and you know how heavy it is.

This is where I don't quite get 100% accuracy. Weigh the one from the heavy set and the one from the light against two neutrals. If it goes down, it's the heavy one and it's heavy, up it's the light, same it's one of the two remaining ones and it's light

 

 

 

Ah, worked it out again rather than trying to remember:

 

 

The key for the right column of my resoning is to measure three and not two in the second step.

Put two from the heavy side along with one from the light in the left pan.

Put one from the light, one from the heavy and one from the four un-weighed (neutral) in the right.

If it tips left it's either the light one in the right or a heavy ones in the left. Weigh the two from the heavy side to find out.

If it tips right, it's the light one in the left or the heavy one in the right. Weigh one against a neutral to find out.

If it stays the same, it's your remaining heavy, or one of the two remaining lights. Weigh the two light suspects to find out.

 

 

Edited September 27, 2011 by Schrödinger's hat

[herbbread]

That's awesome. I had originally come up with a solution that wasn't 100% accurate as well within, but it took me much longer to figure out how to do it with 100% accuracy. Well done!