Consider all two digit natural numbers that remain the same when their digits are interchanged, such as 11, 22 and 33. What is the average of all such two digit numbers?

Introduction:
This question asks for the average of all two digit numbers that remain the same when their digits are interchanged. Such numbers have identical tens and units digits and are examples of simple numerical palindromes in the two digit range.

Given Data / Assumptions:
- We consider all two digit numbers where the tens digit and the units digit are the same. - Examples include 11, 22, 33 and so on up to 99. - We must find the average of all these numbers.

Concept / Approach:
The two digit numbers with identical digits form a simple arithmetic progression. Writing them out explicitly, we can observe the first term, the last term and the common difference. For any arithmetic progression, the average of all the terms is equal to the average of the first and last term, that is (first term + last term) / 2.

Step-by-Step Solution:
Step 1: List the numbers: 11, 22, 33, 44, 55, 66, 77, 88 and 99. Step 2: Confirm that these numbers form an arithmetic progression with first term 11, common difference 11 and last term 99. Step 3: There are 9 such numbers, corresponding to digits from 1 to 9 repeated twice. Step 4: For an arithmetic progression, the average of all terms is (first term + last term) / 2. Step 5: Compute (11 + 99) / 2. Step 6: 11 + 99 = 110. Step 7: 110 / 2 = 55. Step 8: Therefore, the average of all such two digit numbers is 55.

Verification / Alternative Check:
We can also sum all the numbers and divide by 9. Sum = 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99. Pairing terms symmetrically, (11 + 99) + (22 + 88) + (33 + 77) + (44 + 66) + 55 gives (110) + (110) + (110) + (110) + 55 = 440 + 55 = 495. Dividing by 9 gives 495 / 9 = 55, confirming the result.

Why Other Options Are Wrong:
Options 53, 54, 56 and 57 do not match the average computed using either the arithmetic progression property or the explicit sum. They arise from incorrect calculations or misunderstandings of which numbers are included.

Common Pitfalls:
Some students may incorrectly include numbers like 10 or 12 that do not have identical digits, or they might forget one of the valid numbers in the list. Others may misapply the formula for the average of an arithmetic progression. Carefully listing the terms and using the formula (first + last) / 2 avoids these issues.

Final Answer:
The average of all two digit numbers whose digits are the same is 55.