Working backwards – a powerful technique of Problem Solving

Some problems are best tackled with working backwards approach

Working backwards approach

Usually when a person starts solving a problem, she approaches the problem from the start point and takes the first step in the form of her first decision. If she is a cool customer, she stops, evaluates and then only takes the second step. This way she moves from beginning to end of the problem. The desired end of the problem is of course the goal or objective or the solution.

Interestingly, even for mathematical problems, there is sometimes more than one way to reach a solution. For real life problems possibility of multiple paths to solution is more. Not all the paths to the solution though, just like life, are of same quality. Quality of solution can be measured in terms of path traversing time, difficulty and other such costs.

You know, to get something you have to give something. Those are trade-offs. You may follow any path to reach what you think is the solution to a problem, but you may not achieve desired efficiency of solving effort as well as desired quality of solution.
We will always recommend you elegant paths of problem solving. Problem solver's approach will enable you take the elegant path to the desired solution.

Today we will expose you to this aspect of problem solving using a mathematical problem.

Problem 1: Cute little aquarium fishes – how many albino angels did Paroma have to start with?

Paroma has a hobby of culturing aquarium fishes for a number of years. She spawns different varieties of attractive aquarium fishes and sells them to selected customers. Last Monday she evaluated her aquarium situation and decided not to have albino angels at all. Accordingly she first sold half of her albino angels and half an albino angel to Novelty House. Next she could sell half of her remaining albino angels and half an albino angel to Hobby Centre. She found that she had 3 albino angels left. She thought it would be much better to gift these 3 albino angels to her best friend Sohini than to sell them. They were too few for a good sale. It was a good decision and Sohini was actually very elated when she got the three beautiful albino angels as a gift from Paroma.

How many albino angels Paroma had to start with?

Restriction: You can’t use algebra here. That is mathematics. You have to solve the problem as a problem solver using deductive reasoning and at most addition subtraction type of basic mathematical operations.

Think for at least half an hour. Don’t give up. It is an important problem. If you find the elegant solution yourself, not only will you be very happy, but more importantly, the learning will go deep in your problem solving mindset which we call as MPSF (abbreviation of the awkward name of Mental Problem Solving Framework). In simple words, you will learn better.

We will now take a break for you to solve the problem before we return again to discuss the solution and the elegant path to the solution.

By the way, it is not a trick problem at all. It is a very honest kind of problem.

Solution: When you go through the problem, you recognize two barriers to reaching the solution – first: you need to resolve the barrier of half an albino angel; and second: of course you have to reach the solution. Unless you get across the first barrier you can’t proceed to solve the problem at all.

What could the phrase, “half an albino angel” mean? You just can’t cut a living thing into two halves and sell it. Can you? Then you notice that this phrase does not come alone, it is used on two occasions, invariably with the phrases, “half of her albino angels” or “half of her remaining albino angels”.

What can you deduce from this observation?

Given that you can’t cut an albino angel into two halves, your new observation can only mean that halving “her albino angels” or “her remaining albino angels” didn’t result in a whole number and so to get a whole number another virtual “half an albino angel” had to be sold.

This is a problem in the number domain, and being sufficiently educated you immediately understand the root cause of this problem – “her albino angels” and “her remaining albino angels” must have been odd numbers.

This is a key information discovery by the use of deductive reasoning (which includes a bit of mathematical knowledge).

Thus overcoming the first barrier, you continue thinking like a problem solver and decide that you have only one number known, that is, the number of albino angels that were left after two sales. It is 3. You do not have anything else to start your problem solving journey with.

So you decide to start with this end result of 3 numbers of albino angels.

What should you do next? You decide that it would be promising to take the first step as: undo the last sale. Now you are working from back to front. Just imagine how you are moving now along the problem solution path.

The last action of the last sale was to sell half an albino angel. To undo this, you add half an albino angel to the remaining 3 to get 3.5 numbers of albino angels. Next you undo the second action (you are moving from back to front and so, this is the second action to you; while moving forwards it would be the first action of the last sale). The action is: sale of half of her remaining albino angels. To undo, you get back this number which must be equal to what you have now, that is 3.5. Getting back the same number means adding the same number which in turn means doubling the number 3.5. As a result you get 7.

What does this 7 represent? This is the number of albino angels Paroma had after her first sale.

Repeat the same process once more: add half to 7 and then double the result. Oh yes, Paroma had 15 albino angels to start with.

At this point now, you have become a little bit of a problem solver because you have enjoyed the powerful new problem solving technique of working backwards or back to front and also used deductive reasoning.

This being a very powerful approach of problem solving in general, we would like to put it under the class of Problem solving approaches in addition to including it in the class of Problem solving techniques.

If you recall, we mentioned these while discussing Problem Solving Armory or PSA. This special armoury consists of many such classes of weapons for conquering your problems.

Before we forget, your problem solving is not finished yet, as you have to do the testing of your solution working from front to back. Just do it and get the satisfaction of correctly solving the problem. Remember, even if you feel that you have become a problem solver, you must not miss the important step of testing the solution, if it can be tested.e

You have solved the problem using problem solver’s approach, that is, first deductive analysis and then working backwards approach. Otherwise you could have used the normal mathematical approach of algebraic operations. But along that path there would have been no deep understanding of the problem anatomy and no fun at all. That is a mechanical routine path.

Can you now enlighten us with a real life commonly experienced problem where almost all would follow the working backwards approach?

Usually, students who decide to study systematically, make study plan working backwards from the exam date.

Any other problem example comes to your mind? I would leave it to you for the time being. Best wishes.

Read my other blogs on Innovative idea generation and its basic principles and Get smart, get innovative usingTRIZ

Learn problem solving by actually solving a problem

To know how to solve problems, you need to actually solve problems

Learning basics

Just as you can’t learn swimming by reading about swimming, you can’t become a problem solver just by reading books or attending lectures on problem solving. You have to actually solve problems and analyze the problem solving experience for learning to solve further.

It might be that you intend to learn real life problem solving but can’t think of any significant real life problem that you may solve. Learning mathematics or any other subject is much easier! Isn’t it? There are too many problem exercises available in the books for you to solve in a lifetime.

One way to get around this problem is to solve artificial problems that are carefully designed for exercising your general problem solving abilities such as deductive reasoning, problem breakdown, analysis, evaluation criteria formulation, pattern identification and recognition, abstraction, innovation and so on. Once you solve such a problem and analyze the process of your problem solving, you are on the path of assimilating the principles, concepts, techniques and approaches of real life problem solving. This enables you to test your skills as a real life problem solver when you confront such a problem as a DA or a problem owner. This practice is to be continuous as will be your growth in abilities. In a short span of time it is possible for you to be able to confront a problem not in a random manner but with confidence being equipped with new resources for systematically tackling any kind of problem that may come.

It doesn’t mean you will become an expert problem solver in no time, it only means your approach towards problem solving will change from random or heuristic to systematic and analytical which will further result in improved quality of the solutions you will reach.

To be able to acquire problem solver’s abilities you have to solve specially designed problems (and real life problems if you get them) and through the process of solving the problems will acquire what I call the Problem Solving Armory (PSA) resources.

Problem Solving Armory resources

Our Mental Problem Solving Framework (or MPSF) uses a suitable set of Problem Solving Armory (or PSA) resources to solve a particular problem that we face at any point of time. The PSA resources may consist of collections of:

·         Basic problem solving principles: these are abstract principles effective in many problem solving situations. You have been exposed to a few. List of such principles is open-ended and personal. You can form your own set of basic principles that you found effective in solving your problems.

·         TRIZ 40 principles: this is a set of 40 inventive principles primarily meant to be used for generating innovative solutions systematically. These are part of TRIZ, a large system for systematic innovation.

·         Problem Solving Techniques: these are different from principles, as each of these techniques gives you a mechanism for dealing with a particular type of problem situation. A technique is not a principle and consists of steps and processes. Examples are: deductive reasoning, problem breakdown, key information discovery, find the troubleshooter, find the champion, find the expert, pattern identification, enumeration, prototyping and so on. There are many and these are not difficult to understand. Knowing a set of these powerful techniques will automatically increase your problem solving ability.

·         Problem solving tools: these are not the same as techniques. For example, computer spreadsheet software is a powerful tool for making the life of a problem solver more comfortable. Further examples are: Venn diagram, Set theory, Sudoku, Alert mechanisms, Graph theory and so on. Working knowledge of such tools automatically increases problem solving ability.

·         Problem solving methodologies: over time many approaches or systems have been created for problem solving by groups of people working in different domains. All are valuable, generally without conflict with each other and practiced by specific groups of practitioners. When we can learn good things from every domain, why should we limit our learning in only one domain? Methodologies are generally large and complex to understand and apply. Examples are: TRIZ, AHP, lateral thinking, brain storming, Synectics and so on.

·         Problem solver’s skills: if we can identify the required skills, we may improve a set of such more important skill of problem solving through prescribed or our own methods. Some of the skills are: Definition, Prioritization, Abstraction, Analysis, Reasoning, Information discovery, Pattern identification, Innovation and so on.

·         Problem solving approaches: these are the most important components of the all-important PSA. By analyzing the problem at hand, MPSF selects a suitable approach and only then proceeds to go further. Approaches are most important.

If you thought or if I had told you that overnight you may become a problem solver, you thought wrong. It takes a lifetime to become a problem solver. On the other hand it is also possible to start walking the path of a problem solver anytime. More than an innovator, you need to become a self-aware problem solver. 

If you solve well-designed artificial problems, analyzing the process, principles, techniques, methods and approaches used in solving the problems you gradually incorporate these in your mental problem solving framework (or MPSF). This in turn will enable you to apply these when you face a real life problem. Unless you had learned the special approaches and techniques that you have used effectively in solving the academic but well-designed problems, you may still have approached a real life problem solving in a random manner, without any finesse.

These specially designed problems are formulated specifically to bring out one or more than one technique or principle valuable for solving real life problems also.

As a first step we will solve one such problem and analyze our learning.

Four square problem

 

Now you have to answer 4 questions about this square. You will have to be ready and think quickly. Okay?

Question 1: Divide the light grey unshaded area of square A into two equal pieces.

Take a pause and think.

Got the answer? Fine. It’s easy, isn’t it?

Just draw a diagonal line from top right corner of the inner shaded square to the upper right corner of the enveloping square.

Get ready for the second question.

Question 2: Divide the light grey unshaded area of square B into three equal pieces.

Still easier? Should be.