## Find the fake ball in 3 attempts—9 ball weighing puzzle

### Puzzle of 9 balls with 1 fake ball

You have 9 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 8 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:** **30 minutes.**

Give it a good try before going ahead.

#### Bonus puzzle

Can you find all possible ways to solve the puzzle?

*No time limit for this one as well as no solution from us. It's only for you, if you are curious.*

### Solution to the puzzle of finding the fake ball among 9 identical looking balls in 3 weighing

What should be the best plan for first weighing? We can't divide 9 in two equal parts; so obviously,

there has to be three parts: two equal sized sets of balls weighed against each other and a third set of balls left aside.

Next question pops up immediately,

how best 9 is to be split up into 3 parts?

Mentally explore quickly what happens if you keep just 1 ball aside and weigh 4 balls against other 4. In the best case, the scale will be perfectly balanced proving that all 8 balls weighed are good balls and the 1 left aside ball is the fake one. Just weigh next the fake ball with 1 out of 8 good balls and you will know whether the fake ball is lighter or heavier than the good balls.

But if such a good fortune doesn't smile on you, you'll have 8 suspect balls, either the fake among the 4 lighter or among the 4 heavier ones.

This path is not promising to us and we'll follow a **general principle of problem solving** *drawn from experience*, **how best to divide and conquer**,

Divide the enemy in smallest sized groups with group sizes as equal to each other as possible.

All the three group sizes to be smallest, 9 is to be split into three groups each with 3 balls,

$9=3+3+3$.

This strategy also follows **principle of symmetry**,

If you increase the symmetry in the problem by an action, that will be your MOST PROMISING ACTION.

We would straightforward decide to **weigh first,**

3 balls with 3 others,a total of 6 with3 balls kept aside.

The figure below shows the **first weighing combination.**

There can be *three results* of your weighing.

#### 1. First result of 1st weighing—the right pan goes down

**Conclusion:** *All six balls are suspect.* Specifically,

1.1.Either the left side up-goingthree balls contain theorfake lighter ball,

1.2.The right side down-going pancontains thefake heavier ball.

The picture below shows the situation.

#### 2. Second possibility—the left pan goes down:

**Conclusion:** *All six balls are suspect. *Specifically,

2.1.Either the left side down-goingthree balls contain the fake heavier ball, or

2.2.The right side up-going pancontains the fake lighter ball. Essentially,these two results would need exactly similar actions as the results 1.1 and 1.2.

We won't analyze these two results further. *Analyzing results 1.1 and 1.2 should be enough for reaching the solution.*

#### 3. Third possibility—the pans are equally balanced:

**Conclusion:**

The

and,three left aside balls must contain the fake ball

All six balls weighed are good balls.

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

#### Exploring third result: Finding the fake ball among 3 left aside suspect balls in 2 weighing

**Second weighing combination for 3rd result:**

Add 1 good ball from 6 to the three suspect kept aside balls and weigh 2 against 2. Out of 6, five good balls are not used.

The following schematic shows this** weighing combination**. **The suspect balls are shaded orange.**

One side must go down.

#### Result 3.1. When pans are equally balanced in 1st weighing AND side with 1 good ball goes down in 2nd weighing

If *the side with 1 good ball goes down, *

3.1.1.

Eitherthe single suspect ball with the good ball (orange shaded and marked L) is,heavier3.1.2.

Or,thetwo right pan balls (orange shaded and marked R) contain the fakelighterone.

You are not totally sure yet, but *one weighing is still left with you.*

**Third weighing scheme for result 3.1 is,**

Weigh two suspect balls in right pan against each other—Orange shaded R against the other orange shaded R. One of these is the

fake lighter ball.

This is shown in the following schematic.

#### Final Conclusions from 3rd weighing for Result 3.1:

There can only be two possible conclusions,

1. The

pan going up contains the fake lighter ball.2.

If the pans balance,the single suspect ball in left panearlieris thefake heavier ball.

First is shown in the figure left and the second in figure right as below.

#### Result 3.2: When pans are equally balanced in 1st weighing AND pan with 1 good ball goes up

In this case also you will take exactly the same action of weighing the two balls in right pan against each other. Only the results and conclusions will be opposite to the results we had just before.

Now is the time to **act against the first result of 1st weigh.**

#### Find fake ball in 9 balls by 3 weighs puzzle: Exploring 1st result of 1st weigh when left pan with 3 balls went up

The schematic of Result 1 of first weighing is shown below for convenience.

And the **conclusion:**** All six balls are suspect. Specifically,**

1.1.Either the left side up-going three balls contain the fake lighter ball,

1.2.Or, The right side down-going pan contains the fake heavier ball.

All six suspected fake balls are shaded orange with upgoing left pan balls marked L and downcoming right pan balls marked R for ease of later reference.

#### Analysis and weighing decision for the second weighing for two groups of 3 suspect balls each

*We have to think out of the box. The number of suspect balls is large at 6. Only redeeming feature is the knowledge that left pan fake ball must be lighter and right pan fake ball must be heavier.*

First decision taken to** balance the upgoing and downcoming trends partially** is,

To EXCHANGE 1 ball between the two pans.

This is a variation of much used ** base equalization technique** used in solving math problems quickly. Observe that this action balances the asymmetry of nature of ball weights and increases the overall symmetry.

And the second decision taken to **decrease the possibility load of second and third weighing** is,

To KEEP ASIDE 1 ball from each pan.

Dividing 6 suspect balls into three groups each with 2 balls follow our earlier strategy,

This reduces the sizes of three splits of 6 to 2, 2 and 2 keeping the sizes of split groups to the minimum.

The second weighing combination is shown below.

In this second weighing, each pan has exactly same combination of ball types,

One L marked ball that can only be the lighter one if found fake, and one R marked ball that can only be the heavier one if found fake.

#### Results of 2nd weighing for the 1st result of 1st weighing: left pan went up and all 6 balls were suspect

There can be three possibilities,

- Left pan again goes up—
**Conclusion:**Either left pan L ball is the lighter fake, Or, the right pan R ball is the heavier fake, - Left pan comes down this time—
**Conclusion:**Either right pan L ball is the lighter fake, Or, the left pan R ball is the heavier fake, and - The two pans are perfectly balanced—
**Conclusion:**Either the kept aside L ball is the lighter fake, Or, the kept aside R ball is the heavier fake.

Conclusions point out to the same result in all three cases—

one of two L-R pair of balls is the fake.

#### Final 3rd weighing and solution of 9 balls puzzle for 1st result in 1st weighing: left pan went up and all 6 balls were suspect

The last step is easy,

Just weigh a good ball against any of the suspect L-R pair of two balls.

The weighing combination is shown below,

An L-ball is placed again on left pan and a good ball in right pan and last weigh done.

There can only be two results and consequent conclusions:

If left pan goes up the L-ball in it the fake ball, and,

If the two pans balance perfectly, the kept aside R-ball is the fake heavier ball.

Visualization being easy the result graphic is not shown.

**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 kept aside balls at one go*.

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