## Find the fake ball in 3 attempts—8 ball weighing puzzle

You have 8 balls and a two-pan balance with no weights.

All of the balls look exactly the same. Each ball weighs same except for one, which is the fake ball, heavier or lighter than the other 7 balls.

How would you find the fake ball if you are only allowed to weigh the balls 3 times using the balance scale?

**Recommended time:** 15 minutes

Give it a good try before going ahead.

### Solution to the puzzle of finding the fake ball among 8 identical looking balls in 3 weighing

Experienced now, you would straightforward decide to weigh first, 3 balls with 3 others, a total of 6 with 2 balls kept aside.

**Note:** To gain experience in solving ball weighing puzzle **you should start with** the puzzle of **finding the heavier among 8 balls in 2 weighing.**

There can be three results of your weighing,

**First—the right pan goes down: **

**Conclusion:** All six balls are suspect. Either the left side up-going three balls contain the fake lighter ball, or the right side down-going pan contains the fake heavier ball. The picture below shows the situation.

**Second—the left pan goes down:**

**Conclusion:** All six balls are suspect. Either the left side down-going three balls contain the fake heavier ball, or the right side up-going pan contains the fake lighter ball. Essentially, **these first two results would need exactly similar actions.**

**The third—the pans are equally balanced: **

**Conclusion:** the two kept aside balls must contain the fake balls and all six balls weighed are good balls.

Let's take up the **third result first.**

#### Third result: Finding the fake ball among 2 suspect balls in 2 weighing

Just take 1 suspect ball from 2 and weigh it against 1 among 6 good balls. **1 of the two is kept aside.** This is the **second weighing.**

If the pan with the suspect ball goes down or up, the ball in the pan can be identified as the fake ball that is heavier or lighter respectively.

If the two pans are equally balanced, the third kept aside ball must be the fake ball, though you don't know whether it is heavier or lighter.

In the **third weighing against a good ball** you would identify that.

**Question:** There can be another way to find out the fake ball from 2 suspect balls in 2 weighing. Can you say how?

But what would you do in case the two pans are not equally balanced in the first weighing?

#### First and second cases: Analysis and weighing decision for the second weighing for two groups of 3 suspect balls

Dealing with the first and second case of one pan going down and the other going up would be equivalent, and so we would explain the case when the right pan goes down and the left pan goes up.

*We have to think out of the box. The number of suspect balls is large at 6.*

First, mark the three balls going up as UP balls and the other three as DOWN balls. Keep track of the nature of each ball in set of balls. This is important, because you would exchange the balls between the two sets.

As the suspect set of balls is as large as 6 you are aware that special technique is needed here.

**What if we replace two balls from the 3 UP balls with two good balls and keep the two balls aside. Also exchange remaining UP ball with 1 DOWN group of balls.**

What are the ball composition of the two pans now?

The previously UP pan contains 2 good balls and 1 DOWN ball, 2 UP balls are kept aside and the previously DOWN pan contains 2 DOWN balls and 1 UP ball.

Okay, Ready now, you will do the second weighing.

#### Analysis of results of second weighing

There would again be three results,

**First:**the two pans are*equally balanced*.**Conclusion**—defective ball is one of the**two UP balls kept aside in the second weighing and it is lighter.****Second:**The right pan, that went down before,*goes down again.***Conclusion**-the three balls on the left pan cannot contain the culprit as 2 of them are good balls and the third was in the DOWN group in first weighing. The UP ball exchanged and now in right pan cannot also be the culprit as it was in the UP pan in the first weighing. So, you are certain that**two DOWN balls in right pan must contain the heavier fake ball.****Third:**The left pan goes down, that is,**the earlier result is reversed**. Conclusion is clear and simple,**either the lone DOWN ball in the left pan is the heavier fake ball or the lone UP ball in the right pan is the lighter fake ball.**

#### Analytical action for 3rd weighing

For the first two cases when you have either two lighter balls or two heavier balls, action is exactly similar and it is simple.

Just weigh one of the two against the other. If the test is on lighter 2, the ball going up would be the fake lighter ball.

And if the test is on 2 heavier balls, the ball going down would be the heavier fake ball.

In the third case when among two suspects 1 is UP ball and the other DOWN ball, you cannot weigh 1 against the other.

**Think over why.** This is a small exercise for you, if you so wish.

Instead, you would weigh, say the lone UP ball with a good ball keeping aside the other suspect heavier ball. In case of equal balancing, the left out ball is the heavier fake ball. In case of good ball going down, the suspected lighter UP ball must be the fake lighter ball.

And the pan with UP ball cannot go down, isn't it? There can only be two possible results in this weighing.

**Parting Question:** Can you solve the puzzle in any different way?

If you seriously explore to find the answer, you may discover new ways to solve this not so easy puzzle, and also *you would understand why we took a composite action and exchanged and replaced balls at one go*.

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