## The intriguing changing pattern repetition riddle with quick solution

The changing length repetition riddle: What is the 2012th letter in the indefinitely continuing sequence: PATTERNIDOF1STPATTERNIDOF2NDPATTERNIDOF3RD...?

### The Riddle

Find the 2012th character in the following indefinitely continuing repeated pattern:

PATTERNIDOF1STPATTERNIDOF2NDPATTERNIDOF3RDPATTERNIDOF4TH...

### Quick solution to the changing length pattern repetition riddle

The key isn't counting every letter. We need a smarter approach. Like division with a remainder, **we'll drop complete repetitions of the pattern. **

**The complexity** in this case is: **the length of the repeated pattern changes**: for example, the length becomes 14 to 15 from 9th to 10th instance of the pattern.

#### More details of systematically dropping complete instances of the pattern

Each instance initially from 1st to 9th has 14 letters. This changes to 15 characters at the 10th instance ("PATTERNIDOF10TH") and 16 characters at the 100th instance.

By dropping the first 9 instances (14 letters each), we eliminate 126 characters (9 x 14). Dropping the next 90 instances (15 letters each) eliminates 1350 characters (90 x 15) for a total of 1476 characters dropped (126 + 1350) in 99 complete instances.

We need to reach the 2012th letter, so we have 2012 - 1476 = 536 characters remaining.

Next, we drop instances with 16 characters each. How many instances do we need to drop to reach (but not go beyond) the next 536th character?

Dividing 536 by 16 (ignoring the remainder for now) gives us 33 instances. This eliminates 528 characters (33 x 16). We're left with 8 characters.

#### The solution to the changing length repetition riddle

We have dropped 132 complete sequences of the repeated changing length pattern and 8 characters are left to reach the 2012th. The next instance will be the 133rd repeated changing length pattern: PATTERNIDOF133RD. Its 8th character I is the 2012th character in the indefinitely continuing changing length repeated pattern.

**Answer** is I.

**Technique used:** Repeated reduction of characters from the target total characters by dropping complete instances of the pattern till the remaining number of characters is smaller than the next complete instance. *This is the basic steps in a division and getting the all important remainder*.

You may refer to a simpler repetition riddle with a slightly different solution approach:

**Can You Solve the PROBLEMSOLVING Repetition Riddle?**

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