A son and his father go boating upstream on a river. After they have rowed 1 mile upstream, the son notices that the father's hat has fallen into the water. Five minutes later he informs his father, and they immediately turn the boat around and row downstream. Exactly 5 minutes after turning, they reach the starting point and are able to pick up the hat there. What is the speed of the river current in miles per hour?

Introduction / Context:
This boating puzzle involves relative motion between a boat and river current. A hat falls into the river and drifts with the current, while the boat changes direction after some delay and later recovers the hat at the starting point. We must determine the speed of the river current in miles per hour using the timing information.

Given Data / Assumptions:

• The boat is initially moving upstream.
• The hat falls into the river at a point 1 mile upstream from the starting point.
• The son notices the loss immediately as they reach that point.
• They continue upstream for another 5 minutes before turning back.
• After turning, they row downstream for 5 minutes and reach the starting point, where they find the hat.
• The hat drifts with the river current as soon as it falls, with speed equal to the water speed.
• We assume constant boat speed relative to still water and constant river speed.

Concept / Approach:
The crucial observation is that the hat, once in the water, simply drifts with the river. The starting point is 1 mile downstream from where the hat fell. The total time between the hat falling and being retrieved at the starting point is 10 minutes. Therefore, the hat covers 1 mile in 10 minutes, and we can compute the river speed directly without needing the boat speed.

Step-by-Step Solution:
Let the river speed be v_r miles per hour. The hat falls off when the boat is 1 mile upstream from the starting point. From the moment the hat falls, they continue upstream for 5 minutes, then turn and row downstream for another 5 minutes. Total time for the hat drifting from the fall point to the starting point is 5 minutes + 5 minutes = 10 minutes. In that time the hat moves 1 mile downstream purely due to the current. Convert 10 minutes to hours: 10 minutes = 10 / 60 = 1/6 hour. Using distance = speed * time, we have 1 mile = v_r * (1/6). Solve for v_r: v_r = 1 / (1/6) = 6 miles per hour.

Verification / Alternative check:
The actual speed of the boat is not required, because the hat and the starting point are in a simple downstream relationship. As long as the hat arrives at the starting point exactly when the boat does, and the times given match 10 minutes total for the hat's travel, the calculation of 1 mile in 1/6 hour is consistent and sufficient to determine the river speed.

Why Other Options Are Wrong:
River speeds of 4, 2, 5 or 3 miles per hour would mean that in 10 minutes the hat would travel less than or more than 1 mile. For example, at 4 miles per hour it would cover only 4 * (1/6) = 2/3 mile, which would not reach the starting point. Therefore these values contradict the given situation.

Common Pitfalls:
Many learners try to involve the boat speed, upstream speed and downstream speed in detail, which complicates the problem unnecessarily. The key is to focus on the hat, which always moves with the current. Another frequent mistake is to forget to convert minutes to hours when calculating the speed in miles per hour.

Final Answer:
The speed of the river current is 6 miles per hour.