One of 81 balls weighs less than the other 80. Can you produce the ball of different weight in four weighings?

Answer 1

Oh, this is wonderful, and so are multiples of 9 in general. The first comparison (on a balance) is of 3 groups of 27 balls each. One group is withheld. If one group on the balance is lighter, choose it; otherwise choose the one that was withheld. The second comparison is of 3 groups of 9 balls, one group is withheld. If one group on the balance is lighter, choose it; othertwise choose the one that was withheld. The third comparison is of 3 groups of 3 balls each, one group is withheld. If one group of three is lighter, choose it; otherwise choose the one that was withheld. The fourth comparison is of 2 balls, with one withheld. If the lighter ball is on the balance, it will be obvious. If the balls on the scale are the same, the withheld ball is the lighter one.

Note: You must know ahead of time if there is in fact one ball of different weight, and whether the different ball is lighter or heavier; any possibilities could be true and you wouldn't necessarily know it from the group comparisons. Interestingly, if you follow the wrong algorithm, (you are trying to solve for the heavier ball when in fact one ball is lighter, or the other way round) you will occasionally pick the right ball, but for the wrong reason! More often, you will be wrong. In fact, if all the balls are of equal weight, you will conclude otherwiseby following the above algorithm, just based on the fact that someone asked you the question. All I have to do is ask you the question and give you 81 balls of equal weight, and you will always identify a ball (incorrectly) as being of different weight. One solution to this problem is to add a 5th comparison, between your chosen ball and any one of the others that you have already demonstrated to be of equal weight. This should get you to see how important it is to think extremely clearly when developing or using algorithms of any kind. Simply assuming that something is true when you do not have verification or certainty can lead you down a wrong and potentially disastrous path.