How to solve Google's clock hands interview riddle

Summary

This engaging riddle challenges viewers to figure out how many times the hands of a clock overlap in a 24-hour day. The puzzle has stumped many, with common answers ranging from 23 to 24, but the correct answer is 22. The video explores various ways to approach the problem, including algebraic calculations, graphical methods, and intuitive reasoning, all leading to the surprising yet mathematically sound conclusion. The video also reflects on how a large portion of viewers struggled with the puzzle, making it a fun and challenging brain teaser.

Takeaways

• 😀 The riddle asks how many times the hands of a clock overlap in a 24-hour period, and the correct answer is 22.
• 😀 Many people incorrectly believe the answer is 23 or 24, with 46% of respondents choosing 24 in a poll.
• 😀 The hands overlap 11 times in the first 12 hours and another 11 times in the second 12-hour period, for a total of 22.
• 😀 A common misconception is that the hands overlap exactly once every hour, but they actually overlap every 1 hour, 5 minutes, and 27 seconds.
• 😀 The trick lies in the fact that the hands don’t overlap in the 11:00 hour, because the hour hand is moving towards 12, preventing overlap.
• 😀 The solution to the problem involves both visual and algebraic methods to determine the exact times of overlap.
• 😀 The minute hand moves at 6° per minute, while the hour hand moves at 0.5° per minute, creating a relative speed difference.
• 😀 The algebraic approach calculates the time it takes for the minute hand to catch up to the hour hand, resulting in overlaps every 60/11 minutes.
• 😀 A graphical method can also be used, where the intersections of the lines representing the hour and minute hands show the overlap times.
• 😀 The puzzle is often used in technical interviews and is seen as a test of problem-solving ability and mathematical reasoning.

Q & A

How many times do the hands of a clock overlap in a 24-hour period?

The hands of a clock overlap **22 times** in a 24-hour period.

Why is the answer not 24 times?

The answer is not 24 because in the 11:00 hour, the minute hand does not catch up with the hour hand, as the hour hand is already moving toward the 12 o'clock position. This results in only 11 overlaps in each 12-hour period.

What was the most common incorrect answer from the poll on YouTube?

The most common incorrect answer was 24, given by 46% of the respondents.

What is the exact interval between the overlaps of the clock hands?

The hands overlap approximately every **1 hour, 5 minutes, and 27 seconds**.

How can you calculate the time of the overlaps algebraically?

To calculate the overlaps, you use the formula for the rate of the minute and hour hands. The minute hand moves 6° per minute, and the hour hand moves 0.5° per minute. Setting up the equation for the time it takes the minute hand to catch up with the hour hand gives an overlap interval of **60/11 minutes**, or about 65 minutes and 27 seconds.

Why does the problem become difficult for many people?

The problem is tricky because it involves understanding that the hands do not overlap exactly once every hour. The slight movement of the hour hand makes it so that the minute hand must catch up each time, leading to overlaps at intervals slightly longer than 60 minutes.

What is the relationship between the speeds of the hour and minute hands?

The minute hand moves at a rate of **6° per minute**, completing a full 360° rotation every hour. The hour hand moves at a rate of **0.5° per minute**, completing a 360° rotation in 12 hours.

How does the graphical method to solve the problem work?

In the graphical method, you plot the rotation of both the minute and hour hands on a graph. The vertical axis represents the position of the hands (0° to 360°), and the horizontal axis represents time. The points where the graphs intersect represent the times when the hands overlap.

How can the overlap times be calculated using an infinite series?

Using an infinite series, the minute hand must 'chase' the hour hand. The initial gap between the hands is 5 minutes, and each time the minute hand advances, the hour hand moves slightly. This process can be modeled as a geometric series, which ultimately results in an overlap time interval of **5 and 5/11 minutes**.

Why is 12:00 AM considered the first overlap of the day?

12:00 AM is considered the first overlap because it marks the start of the 24-hour period. This is the moment when both the minute and hour hands are at the 12 o'clock position, and they overlap for the first time.