# A Balance, and a Fake Ball among the Good Ones

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Suppose there are many balls. One of these balls is a fake. The fake ball looks exactly the same as all other balls but have a slightly different weight (i.e. lighter or heavier), which can only be identified by using a balance. Here, the balance can only be used to determine whether its left or right side (which may contain one ball or many balls) is heavier, or they have the same weight (i.e. balanced). Suppose the balance can determine weight differences with absolute accuracy, yet it can tell nothing quantitative. On each use of the balance, all balls (i.e. all balls that are going to be put on the left side of balance, and all balls that are going to be put on the right side of balance) must be put instantaneously and simultaneously onto the left and right side of the balance.

To give you some clues: If you have only two chances to use the balance and you know that the fake ball is lighter than other balls, you can determine the fake ball from at most nine balls (i.e. 8 good balls and 1 lighter fake ball). The way is: after labeling all the balls as 1-9, on first use of the balance, you put balls labeled 1, 2, 3 on the left side of the balance and put balls labeled 4, 5, 6 on the right side of the balance. If balanced, you know the lighter fake ball must be within 7, 8, 9. Then, on the second use of the balance, you put 7 on the left and 8 on the right. If balanced, 9 is the fake ball, if not, the ball that is weighted by the balance as lighter (i.e. the “up” side of balance) is the fake ball.

If, however, the first time is not balanced, you know the fake ball must be one of the three that on the up side (amongst the lighter balls). Then the second time would be to determine the light fake ball from three balls, which is exactly the same as determining the fake ball from 7, 8, 9 as shown above.

Questions:

(1) How can you determine the fake ball (don’t know lighter or heavier) from 9 balls (8 good balls & 1 fake ball) and tell whether the fake ball is lighter or heavier, by using the balance three times, given that your first use of the balance is having 1, 2, 3 on the left and 4, 5, 6 on the right.

(2) How can you determine the fake ball (don’t know lighter or heavier) from 12 balls (11 good balls & 1 fake ball) and tell whether the fake ball is lighter or heavier by using the balance three times.

(3) If you have four chances to use the balance, what is the maximum number of balls from which the fake ball (don’t know lighter or heavier) can be identified. In this question, you don’t need to tell whether the fake ball is lighter or heavier in the end. (Although this seems like an open question, you will know that you’ve gotten the right answer if you’re able to find it)

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9 balls start with 1/3 on each well say they are the same then I take the other 3 with an extra one labeled then take one of the remaining ones and check them one on each side

12 balls 6 on each side label heavy side balls and not heavy side balls. then take the remaining 3 and put one away if they are even then it is neither and the one put away is it or it will be one of the two.

it would take me a while with the math I know and I am lazy(I think I may not know how).

but I would drop them and see which has the most impact.

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