Solving Nihaal's Clock Direction Problem

The question asks us to determine the direction of the minute hand after 30 minutes, given the initial time and the direction of the hour hand.

Initial State Analysis (Time is Half Past One)

According to Nihaal's watch, the time is half past one, which means 1:30. Let's analyze the positions of the hands at this time:

• The hour hand is between the numbers 1 and 2. Specifically, at 1:30, the hour hand is exactly halfway between 1 and 2.
• The minute hand points directly at the number 6.

Determining Clock Face Orientation from Hour Hand

The problem states that at 1:30, the hour hand is pointing towards the North-East direction.

Let's consider a standard clock face and its directions:

• 12 is typically North.
• 3 is typically East.
• 6 is typically South.
• 9 is typically West.

At 1:30, the hour hand is halfway between 1 and 2. On a standard clock face, the number 1 is \(30^\circ\) clockwise from 12 (North), and the number 2 is \(60^\circ\) clockwise from 12. The position halfway between 1 and 2 is \(45^\circ\) clockwise from 12. The direction \(45^\circ\) clockwise from North is North-East.

Since the question states that the hour hand at 1:30 points North-East, and the position of the hour hand at 1:30 on a standard clock is indeed \(45^\circ\) clockwise from 12 (which is North-East relative to 12 being North), this confirms that the standard mapping of clock numbers to directions (12=North, 3=East, etc.) holds true for Nihaal's watch.

Initial Direction of Minute Hand (at 1:30)

At 1:30, the minute hand points directly at the number 6. Since the standard clock orientation is confirmed (12 is North, 3 is East, 6 is South, 9 is West), the minute hand at 1:30 is pointing towards the South.

State After 30 Minutes

The question asks for the direction of the minute hand after 30 minutes from 1:30. Adding 30 minutes to 1:30 gives us 2:00.

\(1 \text{ hour } 30 \text{ minutes} + 30 \text{ minutes} = 2 \text{ hours } 00 \text{ minutes}\)

So, the new time is 2:00.

Final Position and Direction of Minute Hand (at 2:00)

At 2:00, the minute hand points directly at the number 12.

Using the confirmed clock face orientation (12 is North, 3 is East, 6 is South, 9 is West), the minute hand pointing at 12 is pointing towards the North direction.

Therefore, after 30 minutes, the minute hand would point towards the North direction.

Revision Table: Clock Directions and Time

Additional Information: Clock Hand Movements

Understanding how clock hands move relative to directions is key in these types of problems.

• A full circle is \(360^\circ\).
• The minute hand moves \(360^\circ\) in 60 minutes, so its speed is \(6^\circ\) per minute.
• The hour hand moves \(360^\circ\) in 12 hours (720 minutes), so its speed is \(0.5^\circ\) per minute.
• The position of the hour hand is affected by the minutes past the hour. At 1:30, it's not exactly on 1, but has moved halfway towards 2.
• Direction problems often test your ability to map the circular clock face onto the standard directional compass (North, South, East, West, North-East, etc.).

In this specific problem, the hour hand's direction at 1:30 helped us confirm the alignment of the clock face with the compass directions, simplifying the determination of the minute hand's final direction.