Understanding the Light Switch Puzzle

The video explores a famous puzzle involving 100 light switches and 100 people. Each person toggles switches based on their number, leading to a pattern where only square-numbered switches remain on. The discussion delves into factors, divisors, and the significance of square numbers, concluding with insights into number theory and highly composite numbers.

What is the initial state of the 100 light switches in the problem?

• Every third switch is on.

• Every alternate switch is on.

• All switches are off.

• All switches are on.

What does the second person do in the sequence of actions?

• Turns on every third switch.

• Turns off every second switch.

• Turns on every switch.

• Turns off every switch.

Why are prime numbers significant in the light switch problem?

• They have an odd number of factors.

• They are switched exactly twice.

• They are never switched.

• They are always on.

What happens to a light switch with an even number of factors?

• It remains on.

• It remains off.

• It is switched on and off an odd number of times.

• It is switched on and off an even number of times.

Which type of numbers have an odd number of factors?

• Prime numbers

• Even numbers

• Square numbers

• Odd numbers

What is the prime factor decomposition of 16?

• 2^4

• 2 x 8

• 3^2

• 2^3 x 3

Why do square numbers have an odd number of factors?

• They are always even.

• They are divisible by 2.

• Their factors come in pairs with one duplicate.

• They have a unique prime factor.

What is a highly composite number?

• A number that is a perfect square.

• A number with only two factors.

• A number with more factors than any smaller number.

• A number that is prime.

Which number is the first to have 12 factors?

• 36

• 48

• 24

• 60

What is the significance of the number 360 in the context of factors?

• It has exactly 10 factors.

• It is a highly composite number.

• It is a perfect square.

• It is a prime number.