Key to the riddle is hidden in the two friends' conversation
Discover the critical information hidden in the two friends' conversation by systematic problem solving—a mix of exhaustive approach and reasoning.
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 didn't tell me enough to figure out their ages.
— My oldest kid is a great sports person.
— Aha! Now I know their ages.
Have you also figured out the ages of the three kids?
Recommended time to solve: 6 minutes.
Solution to the two friends' conversation riddle by systematic problem solving: Thinking aloud (while trying to solve)
The first friend knew the number of passengers in the tramcar, but not you. This seems to be the key problem.
But it must not deter you. You are a problem solver with firm belief that you will solve the puzzle.
First thing you do is to identify what seems to be the most informative sentence in the conversation (after all, key to the solution is in the conversation only):
— 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.
You know that in this statement, the second friend conveyed maximum information to the first.
At this stage, it helps to imagine yourself as the first friend.This is role playing.
Role playing
When you play a role, you would imagine yourself in the role of a specific person acting the way the person would act. It helps you to discover new clues to the problem.
Role playing is an often used technique while solving a problem.
So you will work on the Statement in the same way as the first friend did,
—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.
As a first step, you will form all combinations of factors of 36 and their sum by following a systematic approach.
$1+1+36 = 38$, aha, it is not possible as we've last met about 20 years back,
$1+2+18 = 21$
$1+3+12 = 16$
$1+4+9 = 14$
$1+6+6 = 13$.
Check whether with 1 as the first factor in a sequence of three non-decreasing factors, these four indeed are all the factor combinations.
Principle of exhaustiveness and exhaustive enumeration of possibilities
On occasions, you may need to enumerate all combinations according to a set of given rules. To be absolutely sure that you have indeed enumerated ALL combinations, you must follow a suitable step by step systematic method.
This is called Principle of exhaustiveness, an important component in the skill-set of a problem solver.
To be exhaustive in your enumeration of combinations, you followed a bullet-proof system:
Step 1: Fix 1 as the first of the three sequentially non-decreasing factors,
Step 2: With the first factor fixed, fix the second factor in all non-decreasing variations. The third factor will be the remaining one.
Step 3: After the first factor as 1, start with the first factor as 2 and repeat the steps until you reach the first factor as the largest one in a non-decreasing three factor sequence.
Seems difficult? Well, you have already enumerated all the combinations for the first factor as 1!
Now you start with 2 as the first and smallest of the three factors.
$2+2+9 = 13$
$2+3+6 = 11$
$3+3+4 = 10$.
These are all factor combinations.
Make a quick check to be sure that you have correctly followed your bullet-proof exhaustive system to enumerate the set of 7 factor combinations.
Raising the key question
A problem solver has to raise the most relevant key question after she gets some information.
Next, she tries to get the most reasonable answer to the key question.
When the problem remains unsolved, even after analyzing the answer to the key question, the problem solver must ask the second key question.
Think of this problem-solving process as a repeated series of question, answer and analysis.
Key question followed by answer and analysis of the answer until the problem is solved. Three steps rep[eated are,
- The well-formed key question,
- The most reasonable answer to the key question.
- The result of analyzing the answer to the key question.
This is the never failing natural system of Question answer and analysis (QAA).
Back to the problem, you would now form the most relevant question at this stage,
Even after knowing the product and sum of the ages of the kids, why couldn't the first friend tell the age of the kids?
The one and only reason must be,
A pair of factor combinations must have had the same sum equal to the number of passengers in the tramcar.
That must be the only reason for the first friend saying,
—Got you, but you didn't tell me enough to figure out their ages.
This is an informative statement that we have ignored at first.
What is the pair of combinations with the same sum? The combinations are,
$1+6+6 = 13$
$2+2+9 = 13$.
The key to the solution hides in this pair of factor combinations both with sum 13.
How to break this deadlock and solve the riddle?
Time to form the second key question,
Which answer of the second friend can help to pinpoint a single factor combination out of the two as the solution?
Thinking systematically, you reason out that,
The hidden key must be in an answer referring to "age" of the kids.
After all, the whole thing revolves around age of the kids only!
The breakthrough statement will be the one referring to “age” of the kids indirectly.
This is the use of link-search technique effective in solving logic analysis puzzles.
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 identify the statement that refers to the key parameter at that stage.
In this riddle, age is the key parameter. So you search for the statement that refers to the age of the kids.
It takes just a moment for you to identify "oldest" as the word referring to the age of the kids in the answer of the second friend,
—My oldest kid is a great sports person.
Earlier, this answer seemed totally out of context. But now you would use this information to break the deadlock of a pair of factor combinations.
Using QAA technique, you raise the third important question,
What did the key statement say to the first friend for breaking the deadlock?
It said,
The second friend had ONE "oldest" son.
This is contextual understanding of English language. It gives you the most important information yet.
With a look at the two factor combinations, you find the answer straightaway,
$1+6+6 = 13$
$2+2+9 = 13$.
And you find the answer straightaway.
In the first combination, the ages are 1, 6, 6 with two elder sons, but in second combination 2, 2 and 9, you are looking at the only one oldest son.
The ages of the kids are,
2 years, 2 years and 9 years.
Sum up
What did you do?
You have put yourself in the role of first friend, asking the ages of the kids of second friend.
Next, you acted on the information provided by second friend and went into a deadlock—both you and the first friend. You have used up the first informative statement.
You realize that the reason first friend couldn't decide is the key information to you. This implies a pair of factor combinations with the same sum equal to the number of passengers in the tramcar. That is why the first friend expressed his doubt in his statement.
To break the deadlock of two answers, you have then used link search technique to identify a statement referring to the ages of the kids.
As you look again at the rest of the statements with the specific intention of finding age related words, the word "oldest" catches your eye.
When you examine the two combinations again, you find only one has an oldest kid, and you have found your answer.
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