## Use repetition to discover the key pattern

#### The riddle of PROBLEMSOLVING

What will be in the 2012th position if the following sequence continues indefinitely?

PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING.....

**Recommended time:** 1 minute.

**Hint:** is in the headline above.

### Comment

This problem solving riddle is an MT calendar problem that requires creation and use of a new technique based on the pattern of repetition. The technique will be general enough for adapting it to solve different types of repetition, including Calendar problems.

The approach of identifying key pattern and creating a method to use the pattern effectively is the problem solving approach.

### Solution to the riddle of PROBLEMSOLVING - Discover the key pattern by repeated question, analysis and answer

Writing PROBLEMSOLVING repeatedly and counting the characters to identify the 2012th character is out of question.

It is clear that you have to cut-through the problem using a suitable technique.

The **crucial question** you ask yourself,

In which situation can I

identify a character by its positionin the sequencemost easily and quickly?

**Answer** is immediate,

If I start identifying

from the first letterof justONE sequenceof PROBLEMSOLVING.

In that case you'll know in a few seconds that B is in 4th position or the character in 10th position is L with total confidence.

From this conclusion you realize that for knowing the character in say, 20th position, you don't have to *count the position character by character starting from the very beginning of the repeating sequence,*

You would just DROP

all complete sequencesof PROBLEMSOLVING in 20 characters and start counting and checking in theREMAINING charactersleft. That would be equivalent to the easiest case of counting and checking in justone number PROBLEMSOLVING, complete or incomplete.

Knowing that you have got the key to the solution, you ask the next **crucial question**,

How can I get the

characters remaining after dropping ALL complete sequencesof PROBLEM SOLVING forany given number of characters, say, 2012?

Finally the **Repetition pattern based technique** is complete with the realization that,

You'll get the

remaining charactersas theremainderafterdividing 2012bynumber of characters 14of the sequence PROBLEMSOLVING.

*This essentially is based on the Euclid's division lemma.*

Now you have the solution by applying the method.

#### The solution to the riddle of PROBLEMSOLVING

Divide 2012 by 14 to find the remainder to be 10, and 10th character in one number of PROBLEMSOLVING is L.

**Answer:** The character in 2012th position in the repeated continued sequence of PROBLEMSOLVING is L.

### Key concepts and techniques used

By **repeated question, analysis and answers** you have created and used the **Repetition pattern based technique** based on Euclid's division lemma, to solve the riddle quickly and systematically without any guesses or random trials.

What you have done is actually **Systematic Problem Solving** or just **Problem Solving** in short and *with special meaning*.

You have done a lot,

You have created a fresh Repetition pattern based technique from scratch simply by repeated asking of most relevant questions, analyzing the problem and finding the answer to the question.

This way of problem solving itself is,

The broad

Question, Analysis, Answer techniqueorQAA techniquein short.

Just remember for your continued successful problem solving,

You would be able to solve a problem in an area of knowledge mostly unknown to you if you have access to experts who can provide you with answers to your questions.

This is possible as **QAA technique focuses on broad aspects of the problem** and **not on the details.**

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