Arun believes that his own weight is greater than 65 kg but less than 72 kg. His brother thinks that Arun's weight is greater than 60 kg but less than 70 kg. His mother says that his weight cannot be greater than 68 kg. If all three are correct, what is the average of all the different possible integer values of Arun's weight?

Introduction / Context:
This reasoning question tests the understanding of inequalities and intersection of ranges. Once the possible integer values that satisfy all opinions are identified, the problem reduces to finding their arithmetic mean.

Given Data / Assumptions:
- Arun: weight is greater than 65 kg and less than 72 kg.

- Brother: weight is greater than 60 kg and less than 70 kg.

- Mother: weight cannot be greater than 68 kg, which we interpret as strictly less than 68 kg.

- We consider only integer values of weight in kilograms.

Concept / Approach:
We must find the intersection of three ranges and then calculate the average of the integer values in that intersection. The steps are:
1. Convert each statement into a numerical inequality range.2. Find the common range that satisfies all three.3. List the integer values in that range.4. Compute their arithmetic mean.

Step-by-Step Solution:
Step 1: From Arun, weight is in (65, 72), so possible integer values are 66, 67, 68, 69, 70, 71.Step 2: From the brother, weight is in (60, 70), so possible integer values are 61 to 69 inclusive.Step 3: From the mother, weight is less than 68, so possible integer values are 1 to 67, but we will combine this with the previous limits.Step 4: Intersecting all three conditions gives integer values 66 and 67 only.Step 5: Average of 66 and 67 = (66 + 67) / 2 = 133 / 2 = 66.5 kg.

Verification / Alternative check:
Check each candidate value quickly: 66 and 67 are both greater than 65 and less than 72, greater than 60 and less than 70, and less than 68. No other integer simultaneously satisfies all three statements. Hence the set of possible weights is {66, 67} and the mean of this set is 66.5 kg.

Why Other Options Are Wrong:
- 55.5 kg, 77.5 kg and 88.5 kg lie far outside the plausible weight range implied by the three statements and cannot be averages of valid values.
- Any average involving values larger than or equal to 68 or below 66 would contradict at least one family member's estimate.

Common Pitfalls:
- Misinterpreting the mother's statement as less than or equal to 68, which would incorrectly include 68 as a possible weight.
- Forgetting that the problem asks for integer values of weight and including fractional values in the list.
- Taking the mid point of the combined overall range directly without first finding the intersection.

Final Answer:
The average of all different possible integer values of Arun's weight is 66.5 kg.