Problem solving techniques produce lucid solutions
Some of the approaches that produce startlingly elegant solutions in maths problem solving are,
- Base concept approach: It says approach any problem solving from the most basic concepts possible. All higher level concept structures in any discipline are built on these basic concepts. If you can acquire complete control of these basic concepts, you should be able to solve many of the complex problems using the very basic concepts only and that too more elegantly.
- End state analysis approach: In its many forms, this amazingly powerful problem solving strategy, always looks at a problem analytically and from a completely different perspective to reach a solution along a much simpler and shorter path without fail.
- Working backwards approach: It is a much simpler approach than the other two but nevertheless in special cases it provides the best way to the solution.
- Mathematical reasoning: This is a special form of deductive reasoning coupled with logical reasoning adapted to mathematical domain. When a problem cannot be solved easily by following mathematical steps, often we take resort to this form of mathematical reasoning.
- Inductive reasoning: This follows from inductive principle. Though it is taught as a topic in maths, it has wide scope of application in general.
Let’s take up our problem example. This is reportedly a question asked in an MNC interview.
Coffee with tea or Tea with coffee
In an interview, a member of the interview board asked the interviewee,
Imagine a cup of tea and another cup of coffee with equal amounts in front of you. There is a spoon also on a plate. I take a spoonful of tea from the tea cup and mix it thoroughly with the coffee. Now I take the same spoonful from the coffee cup and mix it back to the tea cup so that volumes of liquids in the two cups again become same.
Now can you say, coffee in the tea cup is more or tea in the coffee cup is more?
The Solution
When I first encountered the problem, I had to imagine myself to be in an interview situation. Offhand I could not decide which should be the solution. But knowing how to solve mixing or alligation problems taught in schools I started in that path with assumed volumes, but very soon realized, it won’t be a valid path to take in an interview where all the board members are looking at me with barely suppressed smiles.
I chose the easy path (after all I was actually not in an interview) and looked at the solution. It was the logic behind the solution that surprised me more than the solution itself. At first, I could not believe the reasoning. How could it be possible? There must be some trick in the statements, I thought! Soon enough though, on reconsidering, I understood the bullet-proof quality of the reasoning and finally believed in it.
I didn’t carry out the mathematical calculation for proof. The reasoning was proof enough for me.
The upshot is,
Unless you believe in a new concept fully, you won’t own it (or perceive it fully) and so you won’t be able to reuse or modify it.
The Reasoning
As the beverage volumes in the two cups were same at the beginning and at the end with two stages of intermixing in between, the amount of coffee in the tea cup is the amount missing from coffee cup which is now replaced by the amount of tea from the tea cup. The two impurity volumes would be same.
The Testing
I couldn’t forget the problem and went on asking alert individuals young and aged. Some hesitated to answer. These were the cautious type. Some though quite emphatically responded, “Oh, the tea in the coffee cup will be more because the mixing process started with the tea cup.” Only one of the whole bunch tested answered correctly but she answered by hunch. When asked about the reasoning, she couldn’t provide any convincingly.
All the while I sensed an underlying disbelief in the logic of the solution when I explained. True, I remembered my doubt when I first saw the solution. The logic dealt in abstraction. For belief in the concept to take solid body, more tangible proof is needed, I thought.
The Mathematical way
Let’s assume the volumes of tea and coffee to start with were both 100ml. Also 10ml of tea was taken to mix it with the coffee in the first mixing. In the second mixing activity then the same volume of 10ml was taken from the cup of coffee thoroughly (and homogeneously) mixed with 10ml tea earlier. This 10ml of mixed beverage was mixed back in the first tea cup to make its liquid volume to 100ml again.
After first mix:
Tea in tea cup = 90ml, Coffee in coffee cup = 100ml, and Tea in coffee cup 10ml. Total liquid volume in coffee cup = 110ml of which (being thoroughly mixed) 110 ml liquid had tea 10ml and coffee 100ml uniformly spread throughout the volume (in mathematics one must think as precisely as possible).
Out of this 110ml, 10ml was taken out and mixed back in the tea cup – 100ml remained in the coffee cup.
Let’s apply the unitary method we learned during our school days.
Amount of tea remaing in coffee cup
Step 1: 110ml coffee mixed with tea contained 10ml of tea
Step 2: So, 1ml coffee mixed with tea contained $\displaystyle\frac{10}{110}ml$ of tea
Step 3: So, 100ml coffee mixed with tea contained $\displaystyle\frac{100\times{10}}{110}ml = \displaystyle\frac{100}{11}ml = 9\displaystyle\frac{1}{11}ml$ of tea.
This amount of tea remained in the coffee cup as impurity.
Amount of coffee in 10ml coffee mixed with tea going to the tea cup
Again we will apply our venerable but often ignored unitary method.
Nearly repeating our first step we took earlier, we state now with confidence
Step 1: 110ml of coffee mixed with tea contained 100ml of coffee
Step 2: So, 1 ml of coffee mixed with tea contained $\displaystyle\frac{100}{110}ml$ of coffee
Step 3: So, 10ml of coffee mixed with tea contained $\displaystyle\frac{10\times{100}}{110}ml = 9\displaystyle\frac{1}{11}ml$ of coffee.
This is the amount of coffee that will be mixed in the tea cup in the second mixing step.
It is equal in volume to the tea in coffee cup.
The extension of the problem
Coming this far I thought, why not extend the mixing process to any number of forth and back mixing!
Instead of only one pair of same amount forward and backward mixing, if we repeat the process say, 100 times, even in that case when the process stops our logical reasoning would still give us the answer in a few seconds – but not the mathematical method. It might take inordinately long time.
However many times you repeat the process, as the starting and ending volumes are same, the impurities in both cups would also be same. The volume of impurities would though differ from one pair of mixing to the next.
In this extended form of the problem, I became fully aware of the power of the problem solver’s approach. Mathematical calculation would be routine, tedious and time consuming while the logical reasoning cuts through all details straight to the heart of the problem and then on to the solution.
A variation of end state analysis approach
Even though the logical reasoning in this case is simple, it is elegant and profound in its efficiency and directness in comparison to conventional method.
It would be so, as it is a variation of the powerful End State analysis approach of problem solving discipline. The logic is based on comparison of detailed final state with the initial state. That’s it.
End state analysis approach is one of the most powerful and efficient real life (and academic life) problem solving resources that we know of. In real life problem solving, many times we use this approach unknowingly and naturally. Among its many variations, this is a specific one and of same power and efficiency as other variations.
This is one occasion that compels us to recommend (as was recommended by a few world renowned mathematicians earlier),
Don’t do routine mathematics, do problem solving.