**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 as shown?

PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING.....

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

Don't stop even if you exceed the recommended time, which is not very important. If you solve the riddle yourself, the insight in problem solving and your increased confidence will be the most important gains.

If you do solve the repetition riddle by finding a useful pattern and using the pattern, **here is the second part of the riddle:**

**Repetition riddle part 2:** Can you use the technique you have discovered to solve any repetition riddle? *I will pose the new challenge to you at the end.*

### Let us concentrate on how to solve the PROBLEMSOLVING repetition riddle

The repetition riddle: "PROBLEMSOLVINGPROBLEMSOLVING..." continued 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. Especially when you need to solve the riddle in only 10 minutes. There must be a **smarter way** to approach this challenge, *and it lies in the nature of repetition itself.*

#### Easy experiments

With no clue to the solution, the *best way to approach is to make an easy experiment.*

Write the sequence just 4 times (4 times is a good way to start experimentation, as in three plus one repetition, it should be possible to discover the all important RULE crucial to the solution):

**PROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVINGPROBLEMSOLVING**

Time for next step.

#### Analysis:

Observe, 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 easyâ€”you have to count just 5 letters.

What about finding the 17th letter? Or, the 45th letter? These are also doable, **but more tedious.**

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

Even now, if you don't have the clue, **make a second experiment:** 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.

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?* Again **answer is** simple: *it is the length of one instance of the repeating pattern PROBLEMSOLVING.*

So you know, to find the 44th letter, you have to count 3x14 = 42 plus 2 letters.

**Now think the opposite way:** (This is the **Other Way Round inventive principle**)

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

Now, finding the answer to the original riddle shouldn't be difficult at all.

#### 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 a very simple 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.....

Recommended time limit: 4 minutes.

### Puzzles you will enjoy

Easy to hard **categorized brain teasers** with **systematic solutions** from SureSolv.com:

**Challenging brain teasers with solutions: Long list.**

And for more puzzles, here are a few other websites with rich collections of easy to hard puzzles:

**BrainDen:**Offers a variety of puzzles, including grid-based puzzles, riddles, and more. They categorize puzzles by difficulty, making it easy to find ones that suit your level**.****Puzzle Baron:**Provides a wide range of puzzles sorted by difficulty.**BrainBashers:**Offers a collection of logic puzzles, brain teasers, and other mind-bending challenges sorted by difficulty.**MathsisFun:**While primarily focused on math, MathsisFun has puzzles suitable for all ages. They range from simple to challenging**.****Riddles, Logic Puzzles, and Brain Teasers.****Best Puzzles - Top 10 Brain Teasers.****Top 150 interview puzzles.****Top 35 Number Puzzles to Keep You Challenged.****20 Hard Riddles for Adults: Best Brain Teasers for Adults.****Top Puzzle-Questions Asked in Business Analyst Interview.****Puzzles Archives****.**

Good luck to your problem-solving!