## Decipher the key and the solution will be yours

### The riddle

Two friends met in a tramcar after years. Their talks went somewhat like this,

— How are you?

— Thank you, I am fine.

— You just got married when we met last about 20 years back. It has been a long time. Any children?

— I have three kids!

— Wow! How old are they?

— Well, if you multiply their ages, you would get 36; but if you add them up, you’d get the number of passengers in this tramcar.

— Got you, but you did not tell me enough to figure out their ages.

— My oldest kid is a great sportsman.

— Aha! Now I know their ages.

Have you also figured out the ages of the three kids?

Let's fix the **recommended time limit at 6 minutes.**

### Thinking aloud (while trying to solve)

The first friend could count the number of passengers in the tramcar, but not you. This seems to be the main problem.

But this must not deter you.

You are a **problem solver with firm belief that you would certainly be able to solve the puzzle.**

#### Problem solver's belief

A problem solver must have complete belief that she can solve the problem in hand.

In this case of the riddle, you know that being a riddle *it has a unique answer* and having this knowledge you go ahead **deeper into the problem** ignoring the apparent initial difficulties altogether.

So you focus your **analytic attention** on the riddle to **identify** first the **seemingly most informative sentence**:

— Well, if you multiply their ages, you would get 36; but if you add them up, you’d get the number of passengers in this tramcar.

There is no doubt that in this statement, the second friend imparted maximum information to the first.

One of the *prime principles of systematic problem solving* is **immediate feasible action**,

Start working on the information you received immediately if you can.

**Imagine that you are the first friend.**

#### Role playing

Imagining yourself time to time in the role of various persons involved in a problem and virtually taking the actions they had taken help to discover new clues.

This is the **often used technique** of **role playing** while solving a problem.

So after hearing the first informative statement, you would quickly form the all possible combinations of factors of 36 in your mind while summing up each combination.

You proceed following a system of yours **that would automatically ensure listing ALL possible combinations** of 3 factors of 36 **Exhaustively.**

$1+1+36 = 38$, aha, it is not possible as we've last met just about 20 years back,

$1+2+18 = 21$

$1+3+12 = 16$

$1+4+9 = 14$

$1+6+6 = 13$.

This is where you stop to be sure that these four are indeed all possible factor combinations with 1 as a factor.

#### Principle of exhaustivity and Exhaustive enumeration of possibilities

Oftentimes you would be called upon to enumerate all possible combinations according to given rules. To be absolutely sure without checking and rechecking, that ALL possible combinations you have indeed enumerated, you must follow a step by step systematic method that suits the case in hand.

This is called **Principle of exhaustivity** in the skillset of a problem solver.

So to be exhaustive in your enumeration of combinations, you followed a** bullet-proof system**:

**Step 1:** Fix 1st of the three sequentially increasing factors as 1,

**Step 2:** With the first factor fixed, fix the second factor in all possible increasing variations. The third would automatically be fixed.

**Step 3:** After exhausting 1st factor as 1 start with 1st factor as 2 and repeat the steps till you reach first factor as the largest one with no more two larger or equal factors.

Seems an involved process? Not at all. You have already exhausted 1st factor as 1 easily.

So now you start with 2 as the first and lowest of the three factors.

$2+2+9 = 13$

$2+3+6 = 11$

$3+3+4 = 10$.

These are all possible factor combinations.

You would nevertheless make a quick check to ensure that you have correctly followed your bullet-proof exhaustive system to enumerate the set of 7 possible factor combinations.

#### Raising the key question

A problem solver has to raise the most important question after she gets some information and analyzes it.

Then she will go ahead in getting the answer to the question.

Even after getting the answer to the key question and

analyzing the answerwhen final solution cannot be reached, the problem solverhas to raise the second key question.

A problem solving process in this way can be thought of as **a series of,**

- well-formed key question,
- answers to the key question, and
- results of analyzing the answer to the key question.

This is the never failing natural system of **Question answer analysis (QAA)**.

So now you would think back to form the most important question at this point.

Why did the first friend couldn't tell the ages of the kids even after knowing the product and sum of their ages?

There **must be one and only one reason**,

Two of the factor combinations must have had sum as the same and equal to the number of passengers in the tramcar.

That can be the only reason for the first friend saying,

— Got you, but you did not tell me enough to figure out their ages.

So this also is an informative statement that we have ignored at first.

Which are the combinations with same sum? These are,

$1+6+6 = 13$

$2+2+9 = 13$.

Because of this same sum the first friend could get close to the answer but finally couldn't be sure of the correct answer.

How to break the deadlock and solve the riddle finally?

Naturally, the key is hidden in the two possible factor combinations sum of both of which is 13.

So you form the second key question,

Which of the other answers of the second friend can help to pinpoint the single factor combination as the solution?

Without going the way of random thinking you reason out that,

The hidden key must lie in an answer of the second friend that refer to

"age"of the kids in some way.

Of course, that must be the only way to home in to the crucial statement that hides the key to the solution.

This again is the often-used problem solving technique in solving logic analysis puzzles, the **link search technique.**

#### Link search technique

When you face a series of information carrying statements and are on the job of trying to analyze the statements together to reach a key conclusion (or simplified problem state),

You would search out the

statement that refers to the key parameterthat you have at this point.

In the riddle, age is the key parameter. So you search for the statement that refers to age of the kids in some way.

With this great illuminating realization, it takes just a moment for you to identify **"oldest"** as the word referring to age of the kids in the answer statement of the second friend,

— My oldest kid is a great sportsman.

In the beginning, this answer seemed totally out of context, isn't it? But now you have only two possible factor combinations on which you have to apply this new information.

So you raise the **third important question**, (following your systematic question forming and answer seeking method faithfully),

What does this statement say to the first friend to break the deadlock?

Yes, of course, it says,

The second friend has ONE oldest son.

This is **contextual understanding of English** and results in the most important information yet.

With this total clarity and control of the situation, you just have a look at the two possible factor combinations,

$1+6+6 = 13$

$2+2+9 = 13$.

The answer is right in front of you now.

In the first combination, the ages are 1, 6, 6 with two elder sons, but in second combination 2, 2 and 9, you have **only one oldest son.**

So this must be the solution by all means. The ages of the kids are,

2 years, 2 years and 9 years.

### Sum up

What did you do?

First you put yourself in the position of the first friend asking the ages of kids of the second friend.

Then you acted immediately on the information that the second friend had provided and got in a deadlock—both you and the first friend. You have used up the first informative statement.

At this point you realize that the reason why first friend couldn't decide is the key information to you. That means there can only be two possible combinations of three factors with same sum equal to the number of passengers in the tramcar.The doubt was expressed by the first friend in the second informative statement and you have used it up.

To break the deadlock of two possible answers, you have then resorted to the use of link search technique to identify a statement that refers to ages of the kids in some way.

As you look again at the rest of the statements with specific intention of finding age related words, the word "oldest" catches your eye. This must be statement that would break the deadlock of two possible combinations.

Now illuminated fully when you examine the two possible combinations again, you find that only one has an oldest kid and you have found your answer.

All through you have followed a systematic problem solving method using a number of important and powerful general problem solving techniques.

This is a little riddle but is a great study in step by step reasoning and decision making—in other words, systematic problem solving.

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