Any more ways to reach the goal?
Some people stop satisfied when they reach a solution of a problem. Some others, the curious ones, go on to explore. Is there any other way to reach the goal? They ask themselves as they move on to find possible many ways to reach the goal state.
In academic problem solving where we usually have one unique solution, very often there are more than one way to reach the solution. It is the job of the explorer to find all the paths to the solution so that the more elegant and cost efficient path can be selected out of the many. Unless you tread all the paths how can you assess which is the best one!
It may seem at first glance a wasteful and time consuming approach with no tangible gain. To an extent it is true. This technique when applied to a particular problem may not produce the elegant desired best path in a short while, as even after reaching the elegant solution you may continue to look for other ways to the solution to verify whether the present path to the solution is indeed the shortest feasible path.
This is why we have made it clear in the beginning that the many ways technique is not to be used for actual problem solving. Rather it is to be practiced for improving your inherent problem solving skill in finding the most efficient path to the solution of a problem. It is a skill improvement technique and is a powerful one at that.
The practical idea is – after you solve a problem and implement the solution, when you get time, analyze and look for other paths to the solution. Try to find all the paths and then compare between the paths to select the best one according to analytically justified criteria.
The action of analysis and finding more paths to the solution itself is a valuable element for improving your inherent option exploration ability. In fact, if you stop to think, in most problem solving situations, a crucial ability is to find the best among many possible paths to the solution.
Practice on academic problems
Real life problems of large variety do not usually occur in our individual lives. On the other hand carefully tailored academic problems can be found aplenty to hone your many ways finding skill. After all, it is a fact that,
The core set of academic problem solving skills itself is used along with a few other cognitive faculties to solve tough real life problems.
Continued practice of applying many ways technique as a habit directly improves your real life problem solving skill also.
Let us elaborate the concept of Many Ways Technique by solving an academic problem.
Measure 4 Litres of water with two empty jugs of capacities 3 Litres and 5 Litres and abundant supply of water.
How would you approach this apparently easy problem?
Usual random approach by trial and error
In about a maximum of 10 to 15 minutes following usual trial and error approach we should be having our first solution.
- Step 1: Fill the 5 Litre jug.
- Step 2: Pour from 5 Litre jug to 3 Litre jug to fill it fully. Status: 5 Litre jug contains 2 Litres of water and 3 Litre jug is full.
- Step 3: Empty 3 Litre jug. Status: 5 Litre jug contains 2 Litres of water and 3 Litre jug is empty.
- Step 4: Pour 2 Litres of water from 5 Litre jug to 3 Litre jug. Status: 5 Litre jug is empty, 3 Litre jug contains 2 Litres of water.
- Step 5: Fill 5 Litre jug fully. Status: 5 Litre jug is full and 3 Litre jug has 2 Litres of water.
- Step 6: Pour water from 5 Litre jug to 3 Litre jug to fill it fully. Status: 5 Litre jug contains 4 Litres of water and 3 Litre jug is full. We have reached the goal.
It took us six steps to reach the stated objective and so the final solution. Nobody told us not to throw out water by emptying a jug. So we have not violated any condition. We are not really sure how we did that, but nevertheless after some mental trials we have got this solution.
Normally we should be satisfied that we have our solution to the given problem. But no, there are more interesting things to go through.
Activity representation – an important problem solving tool
Lots of times we take a series of steps to arrive at our solution. It is important to represent these actions in a systematic and methodical way for,
- Ease of understanding
- Unambiguous and clear visualization of the actions
- Clear situation awareness at each step
- Complete assurance that the solution has indeed been reached, and
- Ease of further comparison and analysis of the method of solution.
If you state your solution in a random manner, chances are that doubts will remain regarding whether your solution is correct or not. Also it won’t be easy to compare multiple methods of solution. Make no mistake - a structured and clear representation of the solution is nearly as important as the solution itself.
The above representation is acceptably well-suited for our problem. Its main characteristics are:
- Only one atomic activity appears at each step. In other words, no step contains two activities.
- The activity or action at each step is represented clearly and unambiguously with no extra baggage of words.
- The actions or events are represented sequentially as they happen in reality. This is important. Good problem solvers have the ability to lay out these actions and events in their mind just as the events may happen in reality.
- Status after taking an action is noted at each step. This makes the situation absolutely clear to the mind and the analyst need to focus only on the last action without burdening the mind with details of previous actions.
Activity representation is an important aid or tool in problem solving. We have used Action schedule to represent the activities here.
Going ahead with our problem, we now wonder – can there be any more solution? Let me make it clear at this point itself – finding out more ways to reach the desired goal of a problem is purely a problem solving skill building technique. If you practice this technique after solving each problem, your skill in finding the most elegant solution is sure to improve manifold.
Try yourself to find a new solution.
Without any more hesitation if we start our effort in finding a new solution to the given problem, in a little while we are sure to land onto our second solution – again by trial and error.
The second solution
- Step 1: Fill up the 3 Litre jug. Status: 5 Litre jug empty, 3 litre jug full.
- Step 2: Pour 3 Litres of water from 3 Litre jug to 5 Litre jug. Status: 3 Litre jug empty, 5 Litre jug contains 3 Litres of water.
- Step 3: Fill up 3 Litre jug again. Status: 3 Litre jug full, 5 Litre jug contains 3 Litres of water.
- Step 4: Fill up 5 Litre jug from 3 Litre jug. Status: 3 Litre jug contains 1Litre water and 5 Litre jug is full.
- Step 5: Empty 5 Litre jug. Status: 5 Litre jug empty, 3 Litre jug contains 1 Litre of water.
- Step 6: Pour 1 Litre water from 3 Litre jug into 5 Litre jug. Status: 3 Litre jug empty, 5 Litre jug contains 1Litre water.
- Step 7: Fill up 3 Litre jug. Status: 3 Litre jug full, 5 Litre jug contains 1Litre water.
- Step 8: Pour 3 Litres of water from 3 Litre jug into 5 Litre jug. Status: 3 Litre jug empty, 5 Litre jug contains 4 Litres of water – we have reached our desired goal.
It’s a great feeling to have another brand new solution. But the nagging question doesn’t leave us – can there be any more solution?
Think before you proceed further.
Now we come to the really interesting part. We have had our first solution, but without stopping at the point, like true blue explorers we went ahead to land another solution in no time. This itself gives us a good feeling of increasing power in finding useful results, not one, but many.
This simple exercise by itself is valuable and in reality strengthens our ability to find more alternatives in a tight situation. The more you practice this technique of finding many ways to solve a problem, better will you be in problem solving in general.
But this is not all. Our focus is not only to find all the alternative paths to the goal. We must also find ways to evaluate which of the multiple solutions is the best one – we need to compare solutions found.
Comparison of solutions
Just by looking at the two solutions we see that the first has six steps and the second has eight. Obviously the first is the better and quicker among the two. But is it enough? Can we think of any more criteria for comparison of the solutions?
Give it a try.
What about wasting water? The first solution wasted 3 Litres of water whereas the second solution wasted 5 Litres by emptying jugs. Again the first solution has come out superior by this new criterion.
If there were more solutions we could have compared all to select the most efficient and quickest one. The actual comparison method may vary from problem to problem. You have to select a suitable one.
Coming this far, we again remember our unanswered question – are there any more solutions?
Need for a systematic approach
Without the help of a systematic approach we can never be sure whether we have indeed found all the solutions. What we really are after is the exhaustive enumeration, and that too following a systematic method.
It is necessary to search for a method to find the best solution among many possible ones as, in general, solution to a problem may be reached via a large number of paths and it might be a very time consuming task to find out all the paths to the solution, compare each pair and then home in to the best one.
The key to this new problem lies in the nature of solution itself. We need to do solution analysis.
Potential solution analysis
This problem of measuring 4 Litres of water using two empty jugs is a small area of analysis. What is the goal here? We have to measure 4 Litres of water by using two empty jugs of capacities 5 Litres and 3 Litres and an abundant supply of water.
If we remove the ideas of “water”, “jugs” and “filling” and apply abstraction technique to the problem, we may very well restate the problem as,
“How can we get a quantity of value 4 by using two values 5 and 3?”
In essence without losing any significant information the original problem can then be transformed into a simple arithmetic problem. This is what we call Domain mapping technique and we have mapped water filling domain to maths domain by using abstraction and keeping only the core elements of the problem.
At the end point 4 Litres of water can be placed only in the 5 Litre jug. This 4 can be arrived at in only three ways,
4 = 5 – 1
4 = 2 + 2
4 = 3 + 1
This represents the exhaustive set of possibilities to reach the end state.
The first possibility represents our first solution and the third possibility our second solution. With a little bit of further thought we can be sure that the second possibility cannot happen with the given empty jugs of 5 Litres and 3 Litres.
Thus our two solutions form the exhaustive set of solutions and the first is the most efficient and quickest path to the desired goal state.
You may wonder now, why did we resort to any trial and error approach at all? We could have started straightaway with this short and quick approach in enumerating exhaustive set of solutions and finally the most efficient solution.
Yes, of course. And the subject of problem solving is about this elegant technique only.
We carried out End State Analysis on the abstracted and domain mapped problem statement to find exhaustive set of solutions and the most efficient one among the solutions. In short we have fulfilled all of our objectives (not only finding a solution) quickly and efficiently using only problem solving armoury resources and techniques along with minimal guessing.
Even if you are not able to resort to such a problem solving approach all through in real life situation, after solving any problem if you do only the exercise of finding more possible solutions, your problem solving skill will automatically be improved.