You have n balls of uniform appearance, and a scale : the scale has one arm, sup

Question

You have n balls of uniform appearance, and a scale : the scale has one arm, supported in the center, with a bowl on each end; you place balls in each bowl, and thescale will either go -up left (U), or -down left (D), or stay- equal (E).

To show that something can be done, you simply give the procedure and argue why it will work in all cases. To show something can NOT be done is harder... you have to either exhaustively survey all possible procedures (which will often be impossible in practice, since there are too many possible procedures...) – or find a feature of the problem that allows you to argue that it is logically impossible to do it.In our case you should exploit the idea that a “weighing” is like a question with a three-way answer (U,D, or E) instead of the usual two – way answer (“yes” or “no”, “true” or “false”, etc.). So, every time you do a weighing, you essentially are reducing the initial uncertainty about what is the case by cutting the “space of all possibilities” into three parts. This idea also gives you a good “heuristic” about what procedures might give the desired result: you will maximize the “information gained” from a weighing by making sure that it will cut the space of possibilities into (approximately) equal parts...!

You have 13 balls, and one of the balls has a deviant weight.

A.Can you locate the deviant ball, AND determine if it is light or heavy, in only three weighings?

B.What if I give you a supply of “normal balls” ( in addition to the 13 that have thesuspect among them). Can you do it now?

Explanation / Answer

A) We can put 6 balls in each bowl. If they weigh equal then the left ball is the deviant ball. If one goes up then we can take the 6 balls with lower weight and divide in the each bowl. the one which goes up has the deviant ball. then we take the balls which contains the deviant ball. Now, we take one ball in each bowl and one ball is left. if it weighs equal then left ball is the deviant ball and if one goes up then that ball is deviant. Thus, we get the deviant ball in three weigh.

B) there would be total 14 balls. We can divide the 14 balls in two bowls (7 each). the one goes up has the deviant ball. now, we take these balls and places three balls in each bowl. Then one ball will be left. If it weighs equal then the left ball is the deviant ball. if one goes up then we take these three balls and place one ball in each bowl. if they weigh equal then the left is the deviant ball and if one goes up then that would be the deviant ball. Thus, we can locate the deviant ball in three weigh.

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