In the arrangement 1 8 5 9 4 7 1 2 5 8 3 6 5 9 2 7 6 4 5 2 9 2 6 4 1 2 3 5 1 4 2 8 3, which digit is fifth to the right of the twelfth digit counted from the right end?

Introduction / Context:
This problem involves positional reasoning within a long string of digits. You are asked to combine two different ways of counting: first from the right end of the arrangement, then from the left, after identifying a particular reference position. Such questions appear frequently in competitive exams and require a clear method for converting positions counted from the right into standard left to right indices.

Given Data / Assumptions:
- The arrangement is: 1 8 5 9 4 7 1 2 5 8 3 6 5 9 2 7 6 4 5 2 9 2 6 4 1 2 3 5 1 4 2 8 3. - We first locate the twelfth digit from the right end. - Then we find the digit that is fifth to the right of that digit (toward the right end). - Positions are counted within one continuous row of digits.

Concept / Approach:
Let N be the total number of digits. The k-th digit from the right is the same as the (N - k + 1)-th digit from the left. After we compute this equivalent left side position, we can easily move “fifth to the right” by adding 5 to the index. This method converts all instructions into one left to right indexing system, which is simpler to manage accurately.

Step-by-Step Solution:
Step 1: Count the total number of digits in the arrangement. Step 2: There are 33 digits in total. Step 3: The twelfth digit from the right is at position N - 12 + 1 from the left, which is 33 - 12 + 1 = 22 from the left. Step 4: Now list the digits with left to right indices around this position: 21: 9, 22: 2, 23: 6, 24: 4, 25: 1, 26: 2, 27: 3, 28: 5, 29: 1, 30: 4, 31: 2, 32: 8, 33: 3. Step 5: At position 22 from the left, the digit is 2. This is the twelfth digit from the right. Step 6: We now need the digit that is fifth to the right of this position. Step 7: Moving five places to the right means adding 5 to the index: 22 + 5 = 27. Step 8: At position 27 in the above list, the digit is 3. Step 9: Therefore, the digit that is fifth to the right of the twelfth digit from the right is 3.

Verification / Alternative check:
We can also verify by counting directly from the right. The twelfth from the right is at index 12 in a right to left list, then moving five places toward the right end corresponds exactly to five additional positions toward the right edge of the original sequence, which is the same as moving to index 17 from the right. Converting index 17 from the right back to a left index also lands on position 27, confirming that the digit there is indeed 3.

Why Other Options Are Wrong:
1: This digit appears nearby but not at the required position after both steps of counting. 2: This is the digit at the reference position (twelfth from the right), not the one five steps further to the right. 7: Although present in the arrangement, it is not located at the specific index calculated.

Common Pitfalls:
A common mistake is to misapply the formula and use N - k instead of N - k + 1 when converting from right side indexing to left side indexing. Another error is to confuse “fifth to the right of” with “fifth from the right,” which are different instructions. Keeping all counts on a single left to right index line after conversion minimizes such confusion.

Final Answer:
The digit that is fifth to the right of the twelfth digit from the right end is 3.