**Repetition riddle:** What is in 2012th position of PROBLEMSOLVINGPROBLEMSOLVING... continued indefinitely? To solve, don't count, find a rule in repetitions.

### The repetition riddle of PROBLEMSOLVING

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

PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING.....

**Recommended time to solve**: 10 minutes.

If you solve the riddle yourself, the insight in problem solving will be the most important gain.

**Bonus puzzle: Second part of the riddle**

**Repetition riddle part 2:** Can you use the technique you have discovered to solve any repetition riddle? *You'll find a new repetition riddle the end.*

### How to solve the PROBLEMSOLVING repetition riddle

- The repetition pattern "PROBLEMSOLVINGPROBLEMSOLVING..." continues indefinitely.
**The tricky question:***What letter occupies the 2012th position in this sequence?* - Counting the letters 2012 times in the endless sequence is out of question. It will be extremely tedious and time-consuming. There must be a
**smarter way**to approach this challenge. **Make an Easy experiment:**With no clue to the solution, the*best way to approach is to make an easy experiment.***Objective:**To find the RULE in the repetition pattern that will be valid for any number of repetitions.

**Experiment:** Consider the pattern repeated four times. It is easy to put down and analyze as well.

- PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING

**Analysis of the 4 times repetition pattern: To know what is easy and what is not**

- The position of say, the 5th letter in the sequence falls under the first instance of the repeating pattern. So finding the 5th letter in the sequence is dead easy—
*you have to count just 5 letters.* **Question:**What about finding the 17th letter? Or, the 45th letter? These are also doable, but more tedious.

#### Finding the Clue to the solution: Asking crucial questions about the difficulties, getting answers and analyzing

**Question is:**How to reduce the number of letters to count?*It will be possible only if you could somehow drop the largest possible chunk of the sequence in the counting process!*There lies the clue to the solution.

**To be more clear,** Count the position of a specific letter R in the first, second, third and fourth instances of the repeating pattern. You have to count respectively, **2, 16, 30, and 44 times.**

**Question:**Any pattern in the four numbers? Oh yes, it is:**the numbers are increasing by exactly 14.****Question:***What is so special about this number 14?***Answer:***it is the length of one instance of the repeating pattern PROBLEMSOLVING.***Nearly there:**So you know, to find the 44th letter, you have to count 3x14 = 42 plus 2 letters.

#### Now think the opposite way: Use Other Way Round inventive principle to find the all important rule in the pattern

**Question:**How to know what letter occupies the 44th position EASILY when you know the second letter is R?**The answer leads to the final clue:***Just reduce 44 by 3x14=42 and count to 2***in one remaining instance of PROBLEMSOLVING.**

*Now, finding the answer to the original riddle should be easy.*

**Hint:** *Divide 2012 by 14 to find the number of complete sequences of the pattern in 2012 characters. Result will be a Quotient (complete sequences) and a remainder (part of the last sequqnce). Safely drop the complete sequences and start counting from the first letter of the remainder of the last incomplete sequence to its end (the 2012th position in the repeated pattern).*

#### Think of the important points:

- When you didn't have any idea how to solve the problem, you have done experiments and analyzed results.
*Commercial research and production environments use*this**Prototyping technique**frequently.- You have formed a leading question based on the results of the experiments, found the answer, analyzed it and formed the next leading question, in a series of such questions, answers, and forming the next question analyzing the answer.
- This is a time-tested
**technique**of**Question, Answer and Analysis (QAA technique)**in a series. - When you reached near the solution, you have
*reversed your line of questioning*: this is the**Other Way Round inventive principle.**

Finally, *you must have used an elementary math concept that most people should know.*

- Now think of forming new types of repetition pattern riddles and solving the problems using the new technique you have just learned, even if you need to modify and adapt the technique.

### A new Challenge for you:

What will be the 2012th letter in the indefinitely continuing sequence:

PATTERNIDOF1STPATTERNIDOF2NDPATTERNIDOF3RD.....

Take your time.

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*Enjoy puzzle solving while learning problem solving techniques.*