In arithmetic reasoning, what is the smallest possible number of ducks that can swim in the described formation where there are two ducks in front of a duck, two ducks behind a duck, and one duck between two ducks?

Introduction / Context:
This puzzle style question about ducks is a classic example of arithmetic reasoning and logical arrangement. The wording sounds as if many ducks are needed, but with careful visualization we can see that a much smaller number satisfies all conditions. Questions like this test the ability to translate a verbal description into a mental picture and to avoid being misled by the apparent complexity of the language.

Given Data / Assumptions:
- Ducks are assumed to stand or swim in a straight line for simplicity.
- The formation must satisfy all three conditions at the same time.
- The conditions are: two ducks in front of a duck, two ducks behind a duck, and one duck between two ducks.
- We are asked for the smallest possible number of ducks that fits all conditions together, not each condition separately with different groups.

Concept / Approach:
The key idea is to identify whether one small group of ducks can simultaneously satisfy all three positional conditions. Instead of first assuming a large number of ducks, we test the smallest possible arrangements and see if there is a way to interpret the phrases from the point of view of different ducks in the line. The phrase two ducks in front of a duck does not require that every duck has two ducks in front, only that there exists at least one duck for which this is true. The same logic applies to two ducks behind a duck and one duck between two ducks.

Step-by-Step Solution:
Step 1: Consider three ducks in a straight line and label them A, B, and C from left to right. Step 2: For duck C, there are ducks A and B in front of it, so we have two ducks in front of a duck satisfied by looking from the position of duck C. Step 3: For duck A, there are ducks B and C behind it, so we have two ducks behind a duck satisfied by looking from the position of duck A. Step 4: For duck B, duck A is in front and duck C is behind, so there is exactly one duck between two ducks, which satisfies the third condition. Step 5: All three conditions are met simultaneously by the same arrangement of only three ducks, so there is no need to add more ducks.

Verification / Alternative check:
We can verify by trying to use fewer than three ducks. With only one duck, none of the conditions about ducks in front or behind can be met. With two ducks, there can be one duck in front of another and one behind another, but we can never have two in front or two behind. For four or more ducks, many arrangements will also satisfy the conditions, but the question specifically asks for the smallest number. Since three ducks already satisfy all statements and fewer than three cannot, three is the minimal valid answer.

Why Other Options Are Wrong:
Option 5: With five ducks the conditions can be satisfied, but this is not the smallest possible number, so it does not answer the question correctly.
Option 7: Seven ducks are more than enough, but again this ignores the requirement to find the minimum number of ducks that works.
Option 9: Nine ducks also can be arranged to meet the description, yet the question clearly emphasises the smallest possible number, which nine is not.
Option None of these: This would mean that no listed number works, but we have already demonstrated that three ducks do work, so this option is incorrect.

Common Pitfalls:
A frequent mistake is to assume that every duck in the group must simultaneously have two ducks in front and two behind, which forces students to imagine long lines of ducks. Another common error is to forget that the conditions can be satisfied by different ducks in the same formation. Learners may also overlook the minimality requirement and choose a larger number that works without checking whether a smaller arrangement is possible. Visualizing the positions or drawing a simple sketch usually reduces these errors.

Final Answer:
The smallest possible number of ducks in the formation is 3.