Unlock Your Inheritance: Cracking the Locker Riddle

Imagine being invited to the reading of your eccentric uncle's will, only to discover a perplexing riddle standing between you and a hefty inheritance. This isn't just any puzzle; it's a challenge designed to test your wits against 99 other relatives, all vying for a piece of the pie. Let's dive into this intriguing scenario and uncover the solution to the locker riddle.

The Inheritance Puzzle

Your uncle, anticipating the potential for family squabbles, devised a clever plan. Instead of directly bequeathing his fortune to you, he presented a riddle that requires collaborative problem-solving. However, a hidden clause offers a tantalizing reward: the first to decipher the pattern and solve the problem without brute force gets the entire inheritance.

The stage is set in a secret room containing 100 lockers, each concealing a single word. The objective? To determine which lockers will remain open after a series of manipulations by you and your relatives.

The Locker Challenge: A Step-by-Step Breakdown

Here's how the challenge unfolds:

• Each relative is assigned a number from 1 to 100.
• Relative #1 opens every locker.
• Relative #2 closes every second locker.
• Relative #3 changes the status of every third locker (opening closed lockers and closing open ones).
• This pattern continues until all 100 relatives have taken their turn.
• The words within the lockers that remain open hold the key to cracking a safe code.

Decoding the Riddle: Finding the Pattern

Before your eager cousins can even begin, you confidently declare that you know which lockers will remain open. What's your secret?

The key lies in understanding the relationship between a locker's number and its factors. The number of times a locker is touched corresponds to the number of factors it possesses.

Consider locker #6 as an example:

• Person 1 opens it.
• Person 2 closes it.
• Person 3 opens it.
• Person 6 closes it.

The factors of 6 (1, 2, 3, and 6) determine the interactions with this locker. A crucial insight emerges: lockers with an even number of factors will ultimately be closed, while those with an odd number of factors will remain open.

The Perfect Square Revelation

Most numbers have an even number of factors because factors typically come in pairs. However, there's an exception: perfect squares. These numbers possess one factor that, when multiplied by itself, equals the number. For instance, let's examine locker #9:

• Person 1 opens it.
• Person 3 closes it.
• Person 9 opens it.

Since 3 x 3 = 9, the factor 3 is only counted once, resulting in an odd number of factors. Therefore, all lockers corresponding to perfect squares will remain open.

Claiming Your Inheritance

You confidently identify the ten lockers representing perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100). Opening them reveals the message: "The code is the first five lockers touched only twice."

This adds another layer to the puzzle. Lockers touched only twice must be prime numbers, as they have only two factors: 1 and themselves. Thus, the code is 2-3-5-7-11.

With the code in hand, you approach the safe and claim your well-deserved inheritance. Your relatives, too preoccupied with their own squabbles, failed to grasp the significance of your uncle's riddles.

The Takeaway

This locker riddle exemplifies the power of pattern recognition and mathematical insight. By understanding the properties of factors and perfect squares, you can solve seemingly complex problems with elegance and efficiency. So, the next time you encounter a challenging puzzle, remember to look for the underlying patterns – they might just lead you to a hidden treasure.