(Story for children) The Math Quiz

When the Great Leader was just 10 years old, he faced a challenging math quiz from his teacher. The quiz presented a series of equations using the same digit three times, student have to add arithmetic operations between the three numbers to make each of them equating to 6, but the solutions seemed impossible using basic arithmetic operations:

The Quiz

0 0 0 = 6
1 1 1 = 6
2 2 2 = 6
3 3 3 = 6
4 4 4 = 6
5 5 5 = 6
6 6 6 = 6
7 7 7 = 6
8 8 8 = 6
9 9 9 = 6

With this quiz, using arithmetic operations (+, −, ×, ÷) makes it impossible to solve all parts of it. The teacher expected students to either solve the equations or indicate when no solution was possible. Here’s her answer sheet:

The "correct" answer sheet

0 0 0 = 6; no solution
1 1 1 = 6; no solution
2+2+2 = 6 or 2*2+2 = 6 or 2+2*2 = 6
3*3−3 = 6
4 4 4 = 6; no solution
5÷5+5 = 6 or 5+5÷5 = 6
6+6−6 = 6 or 6−6+6 = 6
7−7÷7 = 6
8 8 8 = 6; no solution
9 9 9 = 6; no solution

At first, the Great Leader approached the quiz like any curious child, attempting to solve the equations using basic arithmetic operations. However, he soon realized the limitations of simple addition, subtraction, multiplication, and division. When he couldn't find solutions for every equation, he decided to explore deeper. He visited the library and poured over an algebra book meant for intermediate students. With newfound knowledge, he began to discover clever solutions:

With 4 4 4 = 6, the answer is 4+(4^0)+(4^0)=6 as 4^0=1
With 8 8 8 = 6, the answer is 8−(8^0)−(8^0)=6, as 8^0 also = 1
He even found a more 'cool' way to solve 5 5 5 = 6 by using square root: √5*√5+5^0 = 6.

However, he struggled with: 0 0 0 = 6, 1 1 1 = 6 and 9 9 9 = 6.
He borrowed the book home, read clearly again, and realized a key point that not only helped him solve the quiz from 0 to 9, but all real numbers, which are factorial. He realized that with 1 1 1 = 6, the answer is (1+1+1)! = 6 as 3! = 6. But then he realized, "No way, I can make all real numbers become 1 with x^0." With that, he came up with a general formula that states:

"For the quiz x x x = 6, ∀x ∈ ℝ; the answer is [(x^0)+(x^0)+(x^0)]!=6," but wait, something is wrong. Yes, that is 0, as 0^0 is not 1 but undefined. Disappointed, but quickly, he figured out the way to solve 0 0 0 = 6, also using factorials as 0! = 1. So this is his final statement:

The Great Leader's final answer

"For the quiz x x x = 6, ∀x ∈ ℝ \ 0; the answer is [(x^0)+(x^0)+(x^0)]!=6.
For the quiz 0 0 0 = 6; the answer is (0!+0!+0!)!=6"

Despite completing the homework, there was something he also wanted to figure out, which is the imaginary number, number i. But with i^0=1, the answer is simply [(i^0)+(i^0)+(i^0)]!=6.

The following week, he submitted his homework to the teacher. She glanced over it, her expression a mix of confusion and disbelief. Calling the Great Leader to the front, she asked him to explain his work. He calmly detailed his calculations, emphasizing the magic of factorials and the significance of 0!=1 as well as why x^0=1 except 0. His classmates struggled to understand, while the teacher seemed skeptical, having never encountered 0!=1 before. He introduced the algebra book to the class, and the teacher was astonished, wondering how a 10-year-old could grasp such concepts. After a moment of reflection, she made her decision: “0 mark. You may be correct, but this isn’t appropriate for your age.” The Great Leader remained expressionless, returning to his seat as laughter erupted around him. Yet, deep down, he realized he had gained far more than just a grade.