## Solve the repetition pattern riddle by discovering a rule in repetitions

What is in 2012th position of PROBLEMSOLVINGPROBLEMSOLVING... continued indefinitely? To solve the repetition pattern riddle, find a rule in repetitions.

#### The repetition pattern riddle of PROBLEMSOLVING

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

PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING.....

**Recommended time:** 5 minutes.

#### Comment

To solve the riddle, apply a fresh problem solving technique based on the pattern of repetition. It is preferable that the technique is general enough for solving problems with different repetition patterns, including Calendar problems.

### Solution to PROBLEMSOLVING repetition riddle: Find key pattern by repeated question, analysis and answer

Ask the **most important question** (to yourself) and cut-through the problem,

When can I identify a character at a specific position in the sequence most easily?

**Answer** is immediate,

If I check from the

first letter of only ONE sequenceof PROBLEMSOLVING.

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

How to extend this convenient way of knowing the character at a specific position in a repeated sequence of PROBLEMSOLVING?

Let's make a simple trial. To know the character in say, 20th position of such a repeated sequence, what will you do? Will you count and check from the beginning up to the 20th character?

No, you don't need to do so.

Knowing that one instance of PROBLEMSOLVING comprises 14 characters and from 15th character a new instance of PROBLEMSOLVING starts, you will subtract 14 from 20 to get rid of the first PROBLEMSOLVING.

Result 6 will be the effective position of the character you want in one PROBLEMSOLVING. The character will be E.

Another trial. How to know the character in the 32nd position of a continued sequence of PROBLEMSOLVING? Easy. Subtract 2 times 14, or 28 from 32 to get the desired position as 4th from beginning of one PROBLEMSOLVING.

It means you are **dropping all complete sequences of PROBLEMSOLVING** from 32 characters of an indefinitely continued sequence of PROBLEMSOLVING. **The remaining characters are 4.** Check 4th character in PROBLEMSOLVING from start.

*This is the key idea.*

Repeated dropping or subtraction of 14 charactersfrom our target 2012 charactersis equivalent to dividing 2012 by 14.You'll get the

remaining characters as the REMAINDERafter dividing 2012 by the number of characters 14 of the sequence PROBLEMSOLVING.

Euclid's division lemma is the basis of the **repetition pattern based technique.**

#### Final solution to repetition pattern 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 is L.

#### Key concepts and techniques used

By *repeated question, analysis and answers*, a fresh repetition pattern based technique based on Euclid's division lemma created and applied for solving the riddle step by step with no guesses or random trials.

This is the **Question, Analysis, Answer technique or QAA technique in short.**

With access to experts who can answer your probing questions on a specific subject, you can solve a problem in an area of knowledge unknown to you.

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

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