Problem solver’s approach – use the technique of pattern recognition

Use of Pattern recognition technique for problem solving

Not all things have a purpose

Pattern identification and recognition is one of the most basic capabilities of humans (and also of any living thing). Right from birth, the newborn starts identifying patterns using all its senses including visual and auditory. It learns to speak using language and it learns to identify people and objects attaching complex concepts from their visual patterns. Pattern identification goes on throughout our lives. Our learning ability depends on how strong is our ability of identifying new useful patterns.

When a child goes to school first it learns new behavioral patterns to adapt to. When you take up a new position in your workplace, you have to acquire new patterns necessary for functioning effectively in the new position. Sometimes you change track in your career and go into a totally new but promising area. How can you be effective unless you learn all the required concept patterns in your new work domain?

A new Sudoku player starts playing Sudoku for the first time and gets excited. Such a beautiful game! He starts with two Easy level Sudoku problems and quickly reaches the solutions. Immediately he takes up two medium level problems. He finds them a little harder but solves them. How does he do that? He discovers new patterns that are useful for reaching the solution. He does it continuously. Proceeding this way, within a week he might solve the hardest Sudoku problem in the world if his pattern identification mechanism and a few other abilities are strong.  

We use the Pattern identification technique all the time involuntarily and many times voluntarily or consciously in our daily life. When we navigate the city streets by car, we try to identify whether the streets are in a two axis grid or a star shaped formation. When we are to find a particular house in a street we see whether the numbers on one side of the street are even or odd, increasing or decreasing.

Problem 1: Can you calculate it?

Find the unit’s digit of 8²³.

Before going through the solution, please try to solve the problem for about 5 to 10 minutes.

Solution: At first thought you might want to use your calculator, but on second thoughts and being a problem solver, you will start analyzing the problem systematically instead. A problem solver adopts trial and error method, if at all, only as a last resort.

What actions are available to you? You know from experience that no calculator will take such a large calculation. Can you do anything else? Initially you are stuck and can’t think of anything.

But being a problem solver, you examine the problem more closely and realize that full calculation was not wanted at all, but the single unit’s digit was wanted. Now only you have a clear and precise problem definition. This step is very important.  You are actually adopting a problem solver’s approach. You are not trying to solve the problem in any random way.

Now you are more hopeful and think of any other action than actual calculation of 8¹⁹. You remember that when no course of action seems to be visible you can always resort to Enumeration. So you start seeing how the unit’s digit of powers of 8 is affected by increasing the power systematically starting from 1 to 2 then to 3 and so on. This you can certainly do.

A golden principle: what you can do, just do it.

You do the power of 8 calculations starting with power 1 and find that the unit’s digits of 8¹=8, 8²=64, 8³=512, 8⁴=4096, 8⁵=32768, 8⁶=262144, 8⁷=2097152, and 8⁸=16777216 are 8 4 2 6 8 4 2 6.

Do you see anything peculiar in the unit’s digits of the eight calculations? You are surprised to find every 4^th power the unit’s digit repeats in a cycle of 8, 4, 2, and 6. This is the crucial pattern that you have discovered by the use of Pattern identification and recognition principle and Enumeration technique.

Next you apply the Induction principle. Induction principle is taught at school level and not new to you.

As this pattern holds good every 4^th power, it will continue to hold good for every integer power of 8. The power of 23 falls in the 4x5 + 3, that is, in 3^rd place of the cycle and so we will have unit’s digit as 2.

To summarize:

    First you have analyze and find that calculation is not at all possible-the number would be too large. But you also identify that full calculation is not asked for; only the unit’s digit is wanted. Thus you precisely define the problem.

    You think now what course of action is available to you. At least you can start calculating the powers of 8 starting with power 1 and increasing the power one by one. That is enumeration (or prototyping). When no course of action is available, you do enumeration.

    As you can do this, you just do it. You know this golden principle and immediately set out to apply it.

    After a while, you discover the key pattern of unit’s digit 8, 4, 2 and 6 repeating every 4^th power of 8.

    Now you know you are very near to the solution and use the induction principle: as this pattern has repeated every 4^th power for eight numbers of powers (two cycles), it must repeat for all powers of 8.

    Finally, you arrive at the solution: power 23 being 5 full cycles plus 3^rd step of the sixth cycle, the unit’s digit would be the 3^rd number in the cycle 8, 4, 2 and 6, that is, 2.

Pattern matching and corresponding decision making is at the heart of human decision making. In most situations, this process is involuntary, but in many cases, we may be able to reach a just solution by explicitly searching for a pattern in the given problem world.

Most experts work on pattern identification basis and most emergent decision making are basically pattern based and thus intuitive and instantaneous. By working on your pattern identification abilities it should be possible to improve its power and correspondingly your ability to solve real life problems.
 

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