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Solve 3 Puzzles by Solving Only One: Domain Mapping

Solve 3 Puzzles by Solving Only One: Domain Mapping

By solving a puzzle similar to the Generous Gifts puzzle of Dudeney, we'll in effect solve three similar puzzles. This is possible by domain mapping method.

Well, that’s what I will do.

The famed puzzlist Dudeney created his little puzzle Generous Gifts. It reminded me of a Class 10 maths problem that I solved in an unconventional way using Change Analysis Technique first time.

No more digression. Let me first state the Generous Gifts puzzle of Dudeney in essence (not verbatim).

Generous Gifts Puzzle of Dudeney

A rich man used to gift away certain sum of money to the needy applicants each week.

One day he announced, “If there are 5 fewer applicants next week, each will receive 2 dollars more.” But as it happened, next week there were 4 more applicants and each received 1 dollar less.

Question is: how much did each receive in the last distribution?

If you like, you may go ahead and solve this puzzle. I won’t tell you the answer.

But before you try the puzzle you should know about the additional conditions imposed by me first.

Additional conditions on solving the Generous Gifts puzzle of Dudeney

You have to use as little math as possible and try to solve the puzzle within the recommended time of 5 minutes.

“As little math as possible” INVALIDATES the situation where you straightaway form two algebraic equations in two variables and solve. That would purely be a mathematical solution fit for conventional schools.

You have to use the unconventional thinking part of your brain and grapple with the problem using your reasoning.

You may actually take up solving this puzzle, but as for myself, I won’t stop at Dudeney’s puzzle but move on to propose a puzzle I made up exactly similar to Dudeney’s puzzle.

Children in rows and columns puzzle similar to Generous Gifts puzzle of Dudeney

The class teacher in a school one day told one of her colleagues after the school, “If tomorrow students in my class sit in rows with 5 students less in each row than today, there would be 2 rows more.” She continued, “On the other hand, if tomorrow in my class number of students in each row is 4 more, number of rows would be 1 less.”

“Can you tell me what would be the number of students in each row for the second possibility?” She stopped.

As her friend exclaimed, “It’s easy. I’d form two linear equations in two variables for the two possibilities and solve.”

“Oh no, you can’t do that, that would be just a piece of conventional routine mathematical solution. I have not finished yet.” The first teacher interrupted her friend, “There are two more conditions. You have to use as little mathematics as possible and solve the puzzle in 5 minutes. The first condition would make your straightforward conventional algebraic solution invalid. This is a puzzle, you know. This isn’t just a math problem. You have to use reasoning also.”

I would now urge you to solve this version of the puzzle seriously and that too within 5 minutes.

Let me start the solution process.

Solution to the Children in rows and columns puzzle: Domain mapping method helps to form an exactly similar puzzle

First step in solving a problem is always to jot down in precise details what we know about the problem. Especially important are those little pieces of information that are implied but not explicitly mentioned.

Most important pieces of information not explicitly told are,

  1. Total number of students in the class doesn’t change. It is fixed, just as the certain sum of money being distributed in Dudeney puzzle each time.
  2. Each row must have same number of students sitting and each column also would have same number of students just as each applicant receiving same amount of money.

Here the number of applicants is then equivalent to number of columns (or number of students in each row) and number of dollars each applicant receives is equivalent to number of rows.

Now you should be able to see the exact similarity between the two puzzles.

In essence both are same. If you solve one you solve the other.

What is this mechanism? How can the two puzzles that seem to be totally different are IN ESSENCE EXACTLY SAME!

The key to the answer lies in the phrase of “IN ESSENCE”.

This process of reducing a problem or a puzzle to its essential core details is ABSTRACTION. We say, NON-ESSENTIAL DETAILS ARE ABSTRACTED OUT AND ELIMINATED without changing the core of the problem.

But we have taken one more step in forming the second puzzle exactly same as Dudeney’s puzzle.

We have again dressed up the CORE ESSENCE OF THE FIRST PUZZLE in a different dress so that externally it looks totally different from the first.

In other words, in this additional step, we do opposite of REDUCTION BY ABSTRACTION as we did in the first step. We added new non-essential details to the core essence derived in the first step to create a second puzzle that looks different but essential is same as the first.

We call this two-step process as DOMAIN MAPPING.


Following figure gives an idea of this two stage process of domain mapping.

Two stage process of deriving a new problem by domain mapping

The advantage of domain mapping is obvious.

You may know how to solve the students in rows and columns puzzle but may be confused with the Generous Gifts puzzle. Forms of the two are totally different. But if you can spot the exact similarity of the core of the two puzzles, you may solve the first puzzle using your method of solving the second puzzle.

As far as my experience goes,

Domain mapping is one of the most powerful general problem solving techniques.

You may be wondering, “But what about solving the puzzle!”

Just have a bit more patience. I will now do domain mapping for a second time to create a third puzzle exactly equivalent to the first two puzzles.

And instead of solving the first two,

I will solve this third one as I am most comfortable in visualizing the solution of the puzzle in this form.

Solution to the puzzle of changing length and breadth of a rectangle of constant area

The following shows a rectangle of length L metres and breadth B metres that are both integers.

original rectangular area for change analysis

In the first change, length is reduced by 5 metres that increased breadth by 2 metres while keeping the area same. Following shows the effect of this change.

first change in length and breadth of rectangle: change analysis

In the second change, the length increased by 4 metres and to offset the increase in area the breadth reduced by 1 metre thus keeping the area unchanged. Following shows the effect of the change.

second change in length and breadth of rectangle: change analysis

There is no doubt in your mind that to keep the area of the rectangle same, when length is increased, breadth must decrease and vice versa.

You have to find the breadth in the second situation.

In this form of the puzzle, exactly similar to the first two forms of the puzzle, it is easy to understand the basic relationship between the two changing attributes VISUALLY. These attributes are the two variables.

Length in this case is equivalent to the number of applicants in Dudeney's puzzle and breadth is equivalent to dollars each applicant receives in a distribution.

And to fulfill the mandate of solving the puzzle by minimum math, most important is the clear visualization that

Increase in area because of increase in length (or breadth) must be equal to the decrease in area because of the decrease in breadth (or length).

In simpler words,

Change in area due to change in one variable must be same as the change in area due to change in the second variable.

In the powerful problem solving technique of change analysis, we would equate the changes only and apply mathematical reasoning.

Using Problem Solving Technique of Change Analysis

We’d equate the increase in area to decrease in area for both possibilities to derive two equations in two variables that are much simpler in nature and so easier to solve.

Assume original length and breadth of the rectangular area as L metres and B metres.

In the first change:

Increase in area $2(L – 5) = \text{decrease in area }5B$,

Or, $B = \frac{2}{5} ( L – 5) = \frac{2}{5} L – 2$.

Conclusion by mathematical reasoning: For B to be an integer, L must have a factor of 5.

Similarly in the second case,

Increase in area $4 ( B -1 ) = \text{decrease in area }L$.

Conclusion by mathematical reasoning: Again for B to be an integer, L must have a second factor of 4.

It follows,

With two must-have factors of 4 and 5, the lowest value of L is $4\times{5}=20$ metres.

This gives B as 6 metres.

These two values of 20 and 6 satisfy both the equations and so the answer is $(B -1) = 5$. And the total area is LB = 120 sq m.

Following figure shows behavior of the length, breadth and change in area with their actual values in the first change.

Behavior of length breadth and area of rectangle in first change

Even without writing down the pair of equations, it is easy to visualize the relations and reason out the two factors of L.

In fact, once I posed the problem to a young one over phone when he was on the move and he answered correctly in about 15 seconds.

That’s the power of simplified way of thinking.

Last but not the least is the fact that all three puzzles being exactly similar, we have the answer as 5 for all three puzzles.

This is how you can solve 3 puzzles by solving only one of them!

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