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Imagine walking into a room with 100 light switches. They're all flipped to the 'off' position, shrouded in darkness. Now, picture 100 people taking turns toying with these switches, each following a peculiar rule. Sounds like the setup to a brain-bending riddle, right? Welcome to the intriguing world of the 100 Light Switch Puzzle!
This puzzle, sometimes called the 'Locker Problem,' is a favorite in math circles and even pops up in interview questions. It's a delightful blend of simple rules and surprisingly elegant math. Ready to dive in?
The 100 Light Switch Puzzle: How It Works
Here's the gist:
- Person 1: Turns on every light switch.
- Person 2: Turns off every second light switch (2, 4, 6, and so on).
- Person 3: Changes the state of every third light switch. If it's on, they turn it off; if it's off, they turn it on.
- This pattern continues: Person 4 flips every fourth switch, Person 5 flips every fifth switch, and so on, all the way to Person 100.
The Million-Dollar Question: After all 100 people have had their turn, which light switches will be on?
Primes, Factors, and a Dash of Surprise
The initial instinct might be to think about prime numbers. After all, primes have a unique property: they're only divisible by 1 and themselves. While primes play a role in understanding factors, they aren't the key to solving this puzzle.
The real magic lies in understanding factors (or divisors, as some call them). A factor of a number divides evenly into that number. For instance, the factors of 6 are 1, 2, 3, and 6.
Here's the connection to the puzzle: Each time a light switch's number is a factor of the person's number, its state is changed. Let's look at light switch #6:
- Person 1 (factor of 6) turns it on.
- Person 2 (factor of 6) turns it off.
- Person 3 (factor of 6) turns it on.
- Person 6 (factor of 6) turns it off.
Notice something? Light switch #6 gets flipped an even number of times, ending up back in the 'off' position.
The Aha! Moment: It's All About the Squares
The key to cracking the code is realizing that only square numbers have an odd number of factors. Think about it:
- 4: Factors are 1, 2, 4 (odd number)
- 9: Factors are 1, 3, 9 (odd number)
- 16: Factors are 1, 2, 4, 8, 16 (odd number)
Why do squares have this unique property? Factors always come in pairs (1 and the number itself, 2 and half the number, and so on). Square numbers have one factor pair where the numbers are identical (the square root of the number). This 'duplicate' factor is what tips the scales, giving squares an odd number of total factors.
The Illuminating Solution
Since a light switch needs an odd number of flips to stay on, the answer to the 100 Light Switch Puzzle is simple:
The light switches representing the square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) will be on at the end.
Beyond the Puzzle: Why This Matters
The 100 Light Switch Puzzle isn't just a fun brain teaser; it highlights a fundamental concept in number theory. Understanding factors and their properties is crucial in various mathematical fields, from cryptography to computer science.
So, the next time you encounter a seemingly complex problem, remember the light switch puzzle. Sometimes, the most elegant solutions are hidden in plain sight, waiting for you to connect the dots and illuminate the path to understanding.
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