Consider all two digit numbers that remain unchanged when their digits are reversed (for example, 11, 22, ..., 99). What is the average of all such two digit numbers?

Introduction / Context:
This question deals with a special set of two digit numbers: those whose digits are the same and hence remain unchanged when the digits are reversed. Examples include 11, 22, and so on up to 99. We are asked to find the average (arithmetic mean) of all such numbers.

Given Data / Assumptions:

- We consider all two digit numbers with identical tens and units digits.

- These numbers are 11, 22, 33, 44, 55, 66, 77, 88, and 99.

- We must compute the average of this set.

Concept / Approach:
The average of a set of numbers is the sum of the numbers divided by how many numbers there are. For a set of evenly spaced numbers forming an arithmetic progression, the average is also equal to the mean of the first and last numbers. These repeated digit numbers form such an arithmetic progression with common difference 11.

Step-by-Step Solution:
Step 1: List the numbers. The numbers are 11, 22, 33, 44, 55, 66, 77, 88, and 99. Step 2: Recognise the pattern. This is an arithmetic progression with first term 11 and last term 99, and common difference 11. Step 3: Use the formula for the average of an arithmetic progression. The average of an arithmetic progression = (first term + last term) / 2. So average = (11 + 99) / 2. Compute: 11 + 99 = 110. 110 / 2 = 55. Therefore, the average is 55.

Verification / Alternative check:
We can also use the sum and count directly: There are 9 numbers: 11, 22, 33, 44, 55, 66, 77, 88, 99. Sum = 11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99. Pairing 11 with 99, 22 with 88, 33 with 77, and 44 with 66 gives four pairs of sum 110 each, plus 55 in the middle. Total sum = 4 * 110 + 55 = 440 + 55 = 495. Average = 495 / 9 = 55. This confirms our earlier result.

Why Other Options Are Wrong:
- 33, 44, 66: These are individual members of the set but not the average of the entire group. The symmetry of the set about 55 ensures that the average must be 55.

Common Pitfalls:
Some students may miscount the numbers or incorrectly believe the average is the middle term without checking that the sequence is symmetric. Others might attempt to compute the full sum and make arithmetic errors. Using the arithmetic progression average formula is a quick and reliable method here.

Final Answer:
The average of all such two digit numbers is 55.