Just a brain teaser
Part 1: There are 12 balls, one of which is lighter than all the others. Can you find out which ball in 3 weighings?
Divide the 12 balls into 2 groups of six. Weigh to find the lighter six balls. Divide 6 balls into 2 groups of 3. Weigh to find the lighter 3 balls. Weigh two balls - if they are equal, then the third ball is lighter, else it is the lighter of the two.
Part2: Same question as above, except now you dont know if the ball is heavier or lighter.
Divide the 12 balls into 4 groups of three balls each. weigh two groups - thus locate the 6 balls(in two separate groups) which contain the "Ball"; Also note which group goes up and which group goes down in the weighing balance. Weigh one candidate "Ball" group with the known good balls group - this way you find out which group contain the "Ball" - at this point of time, you also know if the group was the "up" group or "down" group, therefore whether the "Ball" was heavier or lighter. Now the three balls can be resolved as above.
My take on this:
Part3: Same question as Part2, except can you do with an initial group of 4+4+4. i.e. in the first weighing, you are forced to weigh 4 vs 4 balls.
I believe this is a non-trivial solution to this and dunno if anyone else has attempted it.
Weigh 2 groups-of-4. If they are both equal, then the "Ball" is in the third group. This can no be made a 6 ball group by adding 2 good balls. The solution now follows Part2 above.
If they are unequal - note which is the "up" group and which is the "down" group. Here's the complicating part:
**** ???? <- let these be the 2 groups of balls
remove 2 balls from group-1 and 1 ball from group-2. So,
** ??? **?
Now there are group-1, group-2, group-3.
Put 2 balls from group-2 into group-1 and pad group-2 with 3 good balls.
**?? ?000 **?
Weigh group-1 and group-2. If the direction of "up" and "down" has changed from previous weighing, then the fault is because of the two ?? balls transferred from group-2 to group-1. If the directions are maintained then the fault is because of the two ** balls in group-1 and ? ball in group-2. Thus now u have 3 balls - **?
If the weighing is equal, then the problem is in group-3, again ending up with 3 balls - **?
Now, we already know that * balls and ? balls belong to one of "up" or "down" group.
From the 3 balls remaining, take one * ball out, and club the remaining * and ? balls together. So,
Weigh the *? ball with two good balls. If the *? goes in the direction of *, or in the direction of ? ("up" or "down"), we know whose fault is it. If they are both equal, the remaining * ball is the culprit.
The interesting thing is in the last possibility - we will never know if the ball was heavier or lighter!
You heard it here first!!!