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