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

Simple Logic analysis is an important resource for Real life Innovation and Problem Solving

Logic analysis technique

Logic analysis

We frequently use the term ‘logic’. People who use logic to back up their comments, decisions or actions are considered as logical. Sometimes this term is interchangeably used with the term ‘rational’. People who do not behave or act according to commonly accepted norms are termed as irrational. Human beings do not always follow stringent logic.

Nevertheless ‘logic’ is a well-defined technical term; there is nothing vague in ‘logic’. In logic domain, there is truth or falsity of propositions, nothing in between. It is kind of black and white Boolean world. One says, the statement or the proposition – “Rainee is a good girl” can either be true or false. Thus this statement or proposition is a valid proposition in logic. “Rainee is a somewhat good girl” in contrast is not a valid proposition because somewhat introduces uncertainty and it is difficult to determine whether the statement is purely true or surely false.

Though real life cannot really be represented or explained using logic, still logic analysis as a technique in real life problem solving undoubtedly is valuable.

Let us look at this concept domain more closely by solving a problem from Martin Gardner’s South Sea adventure stories.

Problem 1: Which path would you take? (version 1)

A logician while visiting the South Seas is at a fork. He needed to know which of the two roads in front of him leads to the village. Perchance he has two willing natives available near at hand. But one of them is from a tribe of invariable truth-tellers, and the other from a tribe of invariable liars. He knows this predicament but does not know which one is the liar or the truth-teller. The natives though know each other well. The logician can ask only one question to any one of the natives. Can he know the right path to the village?

To remind you, Logic Analysis is a discipline where there are statements or propositions which can only be true or false (yes or no) and logicians are experts in dealing with problems in this domain. As you are not a logician, logic domain is new to you. The process of solving logic problems is new to you. But even though you are not a logician, at least you are a problem solver who is supposed to proceed solving any problem systematically.

So you need to proceed using deductive reasoning systematically, not in any random or chaotic manner. Here you must stop and try to solve the problem yourself. You should try at least for 15 minutes for solving the problem. To give you an opportunity of solving the problem yourself, we would delay explaining the solution only a little bit.

Our break is over. Here goes our reasoning.

The logician can ask the all important single question to only one of the natives without knowing whether the native is the liar or the truth teller. Depending on the answer he will know which the right path is. If the answer is different from the two natives he can never be certain. So a necessary condition for the question to surely lead you to the right path, the answer must be same whether the native to whom the question is asked is the liar or the truth teller. How to form such a question?

Consider that a habitual truth teller can be symbolized as a logic transformation box that outputs the same logic variable state it takes as inputs. The following figure depicts the situation,

Similarly, the logic transformation box for the habitual liar will be to reverse the state of input. In other words, if the input is YES, it will output with answer NO and vice versa. The following figure depicts the situation,

So you understand that a single question will surely produce two values as answer from the two natives. This cannot be the solution then. It implies surely that a single simple question will not be the answer-you need then to ask not a simple question but a compound question comprising of two simple questions. You arrive at this conclusion by your deductive reasoning mechanism.

Knowing that reversing twice restores the original state, we may think of asking the habitual liar two questions in conjunction, “Will you answer YES if I ask you whether the path on the right is the path to the village?” The second part of the compound question goes as the first input to the logic transformation box of the habitual liar reversing the state of the input. In terms of our problem it means, if in reality the path on the right were the path to the village, the TRUE answer to this part of the question would be YES, but as the liar must reverse the state, he must answer with NO. This is hypothetical, but still the habitual liar has no other option than to imagine that his answer to the second part of the question would be a NO. This NO then goes as input to the logic transformation box of the liar for answering the first part of the question and completing his whole task.  Being a logic state reversing mechanism, the liar would have to reverse this NO to YES and actually arriving at the TRUE answer to the original core statement. The truth teller poses no problem as he does not do any transformation in each of the two transformation stages. An input of YES passes straight through as YES in the output. Note that, the question could have been “path on the left” instead of “path on the right” and also “NO” instead of “YES”. Try these out. The figure below depicts the transformation sequence for the habitual liar.

Even though you are not a logician, you proceeded by adopting a systematic approach and finally arrived at the solution.

We will learn to solve a more complicated logic problem next session.

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