Working backwards approach

Sometimes you need to start your analysis from the end

Working backwards approach

Usually when we face a problem that involves a sequence of events, we start the process with our first decision regarding the first event we see in front of us, then take the second decision and the next and thus move towards the solution event after event. If we are fortunate, we reach the solution somehow. This is START to END step by step problem solving in forward direction. This also is the naturally adopted direction of problem solving for most of us.

Many a time though we need to start our analysis from the end event and work backwards to the beginning event.

Case example 1: Journey planning

When my niece told me her important admission test will be held at Kalyani in the morning I sensed a problem. Kalyani is a few hours' drive away from Kolkata by road. Morning time meant we have to start early and be careful in deciding how we travel. A wrong decision might lead us to failure in reaching at right time.

By the first hand information it seemed it is faster to reach by train rather than by road. But there may be a number of trains in the morning for Kalyani. Now the decision to be taken is - which train to catch?

I asked my niece, "What is the time of reporting?" She said, not very enthusiastically, "It is 8.45 in the morning. It'd take at least an hour to reach Kalyani station from Bidhannagar (our station). There are buses and other conveyances from the station to the venue of test. It'd take at least 15 minutes more to reach the campus." I told her to decide which train to take and its time of arrival at our station.

Next day when I asked her about the situation, she said a little hesitantly, "I think we should take the 6.15 train." When I asked her how she arrived at the decision, she explained, "I got the times of the morning trains. The journey would take about an hour and fifteen minutes. So we would reach Kalyani at about 7.45 safely if we take the 6.15 train." "Alternatives?", I pursued. "There are a few other trains around that time, but this looked to me the most suitable." She replied. Then I explained the goodness of Working backwards approach for this decision making problem. This is a simple problem, but when we face this kind of problem, we always work backward to be absolutely clear about our first decision.

End state: To report at the venue by 8.45 am. So we must target to reach the venue by 8.30 am latest. 15 minutes is the buffer time. In our daily life to be sure of reaching a place or meeting a deadline we always use this buffer time concept. This caters for unforeseen situations.

End minus 1 step: To reach the gate of the venue by 8.20 am, as at least 10 minutes may be required to reach the test hall from the gate of the large campus. This is a small detail and never to be ignored.

End minus 2 step: We should keep at least 30 minutes to reach the test campus gate from Kalyani station. Kalyani is not a large city and transport options may not be many. You need also to account for the time to get down from the train and select and board the conveyance that would take you to the campus gate. Rolling back 30 minutes from 8.20 am, we should then take such a train that would certainly reach Kalyani by 7.50 am.

End minus 3 step: With this theoretically safe time of reaching Kalyani station, we must add some buffer as the train might delay on the way. It might also arrive at Bidhannagar, our starting station later than scheduled time. These delays happen. So, I concluded, It is safer to choose an earlier train. The choice of  6.15 train seemed to be ok, but we chose the 6.10 train to keep the buffer for canceled or delayed trains.

Not very surprisingly our train arrived a bit late. Getting a conveyance from Kalyani station to the test venue campus was a hassle and we had to take a quick last minute decision. We reached finally by 8.35 - safe and dry.

End minus 4 step: With (End minus 3 step) decision taken, journey planning was not over though. I asked her, "To catch the train arriving at 6.10 am, when should we start from home?" That was not complicated, but still it is better to fix this predictable and more certain event also on the event timeline to make the whole timeline transparent involving least amount of decision making and anxiety during the actual journey.

Practical experience says,

There are always deviations from what you plan and what actually happen.

Though the instinctive decision of my niece was not much off the mark, when we finished the systematic journey planning process using working backwards approach, our decision became more robust and we became aware of all the events in the whole journey that we would have to go through. This awareness of details is useful for tackling unforeseen situations with more confidence. All in all, the whole process increased our confidence in our decisions and resulted in reduced anxiety.

For my niece it was her first journey outside Kolkata where she had to take part in decision making. When I discussed this problem with one of my younger but more experienced friends, his time estimates were spot on and coincided with our estimates. Invariably he started from End event - the goal. My friend remarked casually, "When we started our career we also tried to plan our journey from the starting time, not the ending target time. Now with experience we do backwards journey planning using buffer times automatically."

Experience builds similar kinds of problem solving mechanisms in most people but with varying degrees of intensity.

Working backwards approach is a powerful problem solving technique used in many real life problem solving situations. For planning and execution of large and complex projects with fixed target time point, working backwards is an essentiality.

Though we use this technique frequently in real life, we find from the literature available that formalism to this approach was given by the mathematicians. This is not very surprising, as effects of problem solving techniques or approaches can be very clearly demonstrated in the more certain and specific world of mathematics.

Case example 2: How many angel fishes did Paroma have to start with?

Problem description

Paroma breeds aquarium fishes as hobby and a means of earning too. Once she had certain number of angel fishes. Last Monday she evaluated her fish stock and decided not to have angels at all. Accordingly she first sold half of her angels and half an angel to Novelty House. Then she could sell half of her remaining angels and half an angel to Hobby Centre. She found that she had 3 angels left. She thought it would be much better to gift these 3 angels to her best friend Sohini than to sell them. The number was too few for a good sale. It was a good decision and Sohini was very happy to get the three beautiful angels as a gift from Paroma.

How many angels did Paroma have to start with?

Restriction: We the common folks don't know any mathematics such as linear equations or variables and so we solve problems using mainly our wits. Things like simple addition subtraction division multiplication all of us know. No, sorry, algebra can't be used here. That is mathematics. By the way, it is not a trick problem at all. It is a very honest kind of problem.

Recommended time to solve this problem is 15 minutes. Please give it a try. Unless one tries to solve a problem, one can't appreciate the finer points of the solution. Furthermore, the special problem solving techniques, tools or approaches used for solving academic problems can be used very effectively in solving many important real life problems.


When you analyse this problem, you recognize two barriers to reaching the solution – first: you need to resolve the barrier of half an angel - how can a living fish be halved, you may wonder! Second: if the first bariier is crossed, you need to reach the problem solution. Unless you get across the first barrier you can’t proceed to solve the problem at all.

What could the phrase, “half an 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 occur alone, it is used on two occasions, invariably with the phrases, “half of her angels” or “half of her remaining angels”.

What can you deduce from this observation?

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

This is a problem in the number domain, and being sufficiently aware you immediately understand the root cause of this problem – “her albino angels” must have been odd numbers. These numbers when halved always gave rise to a whole number and a half.

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 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 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 angel. To undo this, you add half an angel to the remaining 3 to get 3.5 numbers of 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 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 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.

If you recall, use of such an approach in solving academic or real life problems, increases your belief on the approach. Additionally if you yourself apply working backwards approach for solving one of your problems, the powerful approach comes closer to being a part of your mental problem solving framework.

Can you identify other real life scenarios where working backwards produces much better results?

Appearing for an exam, preparing for a lecture, organizing for an event where you have preparation time are only some of the real life scenarios where you would automatically use this powerful problem solving approach.