Imagine receiving news that your wealthy, eccentric uncle has passed away. You, along with 99 other relatives, are summoned to hear the reading of his will. Your uncle, known for his love of riddles, has devised a clever puzzle to determine who will inherit his fortune. His intention was to leave everything to you, but he anticipated that your relatives would never leave you in peace if he did so outright. Thus, he crafted a challenge, relying on the riddle-solving skills he taught you.
In his will, your uncle left a note: “I have created a puzzle. If all 100 of you answer it together, you will share the money evenly. However, if you are the first to find the pattern and solve the problem without going through all of the legwork, you will get the entire inheritance all to yourself. Good luck.”
The lawyer escorts you and your relatives to a secret room in the mansion, which contains 100 lockers, each concealing a single word. He explains the rules: Each relative is assigned a number from 1 to 100. Heir 1 will open every locker. Heir 2 will then close every second locker. Heir 3 will change the status of every third locker, specifically opening it if it’s closed and closing it if it’s open. This pattern continues until all 100 relatives have had their turn. The words in the lockers that remain open at the end will help you crack the code for the safe.
Before anyone can begin, you step forward, confident that you know which lockers will remain open. But how did you figure it out?
The key lies in understanding that the number of times a locker is touched corresponds to the number of factors of its locker number. For instance, locker #6 is touched by Person 1 (opened), Person 2 (closed), Person 3 (opened), and Person 6 (closed). The numbers 1, 2, 3, and 6 are the factors of 6. A locker with an even number of factors will remain closed, while one with an odd number of factors will remain open.
Most lockers have an even number of factors because factors typically pair up. However, lockers that are perfect squares have an odd number of factors, as one factor is repeated (e.g., 3 x 3 = 9, but the factor 3 is counted only once). Therefore, every locker that is a perfect square will remain open.
Recognizing this, you quickly identify the ten lockers that are perfect squares and open them to reveal the words inside: “The code is the first five lockers touched only twice.”
You realize that the only lockers touched twice are those corresponding to prime numbers, as they have exactly two factors: 1 and themselves. Thus, the code is 2-3-5-7-11. Armed with this knowledge, you approach the safe, enter the code, and claim your inheritance.
It’s unfortunate for your relatives, who were too preoccupied with their own squabbles to appreciate your uncle’s riddles. Thanks to your keen understanding of numbers, you alone unlock the fortune.
Imagine you are one of the 100 relatives. Create a chart with numbers 1 to 100. Simulate the process of opening and closing lockers as described in the article. Identify which lockers remain open at the end and explain why.
Choose any number between 1 and 100. List all its factors and determine if it has an odd or even number of factors. Discuss why lockers with an odd number of factors remain open and relate this to perfect squares.
Write down the numbers 1 to 100. Identify and circle all the prime numbers. Explain why these numbers are significant in the context of the riddle and how they relate to the code for the safe.
Using the concept of factors and prime numbers, create your own riddle involving a series of lockers or boxes. Challenge your classmates to solve it and explain the reasoning behind your puzzle.
List all the perfect squares between 1 and 100. Explain why these numbers have an odd number of factors and how this knowledge helped solve the riddle in the article. Create a visual representation to illustrate your findings.
Inheritance – In mathematics, inheritance refers to the way certain properties or characteristics are passed down from one generation to the next, often used in algebraic expressions and functions. – In algebra, we can think of inheritance as how a function can take on properties from its parent function.
Riddle – A riddle is a type of puzzle or problem that requires creative thinking to solve, often involving numbers or mathematical concepts. – The math teacher presented a riddle: “I am an odd number. Take away one letter, and I become even. What number am I?”
Pattern – A pattern is a repeated or predictable sequence of numbers or shapes that follows a specific rule. – The sequence 2, 4, 6, 8 is an example of a pattern where each number increases by 2.
Factors – Factors are numbers that can be multiplied together to get another number. – The factors of 12 are 1, 2, 3, 4, 6, and 12 because they can all divide 12 without leaving a remainder.
Perfect – In mathematics, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding itself. – The number 6 is a perfect number because its factors, 1, 2, and 3, add up to 6.
Squares – Squares are numbers that are the product of a number multiplied by itself. – The square of 5 is 25, because 5 times 5 equals 25.
Lockers – In a mathematical context, lockers can refer to problems involving combinations and permutations, often illustrated with scenarios like locker arrangements. – If there are 10 lockers and each locker can be either open or closed, how many different combinations of open and closed lockers are possible?
Prime – A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. – The number 7 is a prime number because it can only be divided by 1 and 7 without a remainder.
Code – In mathematics, a code can refer to a system of symbols or numbers used to represent information, often used in cryptography. – The teacher asked the students to create a code using numbers to represent letters in the alphabet.
Numbers – Numbers are mathematical objects used to count, measure, and label, and they can be whole numbers, fractions, or decimals. – The numbers 1, 2, 3, and 4 are the first four whole numbers in the counting sequence.
