Symmetry, Asymmetry

From time to time, late at night, YouTube serves up intriguing content designed to test your math skills. It probably appeals to our intellectual vanity—we want to prove we're well-educated and can solve these kinds of puzzles with ease. 😄

This time, I encountered this problem. I thought it was a very well-designed question that touches on many crucial mathematical concepts, particularly symmetry and asymmetry with a bit tweak.

This problem can be solved using algebraic substitution. The key insight is to recognize the symmetric structure and use substitution (a.k.a re-parameterization in other fields) to simplify it.

Method 1: Centering Around the Average
Looking at the factors $(x+9)(x+7)(x+5)(x+3) = 9$, we notice they form an arithmetic sequence with average $x+6$. Let $y = x + 6$. The equation transforms to: $(y + 3)(y + 1)(y - 1)(y - 3) = 9$. Then, grouping symmetric pairs: $(y^2 - 9)(y^2 - 1) = 9$. Then, substituting $z = y^2$ gives us the quadratic $(z - 9)(z - 1) = 9$, which simplifies to $z^2 - 10z = 0$.

Method 2: Pairing Inner and Outer Terms
We can also group the original factors as: $(x + 9)(x + 3) = x^2 + 12x + 27$ and $(x + 7)(x + 5) = x^2 + 12x + 35$. Then, the equation becomes $(x^2 + 12x + 27)(x^2 + 12x + 35) = 9$. Let $u = x^2 + 12x + 31$ (the midpoint). Then $(u - 4)(u + 4) = 9$, giving us $u^2 = 25$.

Both methods are essentially equivalent and exploit the symmetric structure to reduce the quartic equation to a simpler quadratic form. This algebraic manipulation technique is the core concept being tested, particularly when direct expansion is impractical or prohibited.

However, adding an integer constraint $x \in \mathbb{Z}$ would completely change the nature of the problem, shifting it from pure algebra into the number theory domain covering factorization. This will be so much fun!

In factorization or prime factorization, there is typically no substantially known symmetry (though there are some such as quadratic reciprocity). This makes factorization a computationally difficult problem. In this question, since we are dealing with the integer 9, the search space is relatively small: $(-1) \times (-9)$, $(-3) \times (-3)$, $3 \times 3$, $1 \times 9$. By further reducing using the monotonic increasing pattern in the question, we can conclude that the solution is $x = -6$ for the integer value.

You might have already recognized that I simplified the process of finding a solution using possible factor combinations. If you work through this process, you'll instantly realize how messy it becomes even with this limited search space. We're not Ramanujan, after all. This illustrates why factorization is computationally hard, which is precisely what makes modern cryptography work - it relies on this computational asymmetry between multiplication and factorization. However, cryptography also relies on concepts of mathematical symmetry.

This might seem perplexing and contradictory! In another post, I'll demystify this when I explain how the RSA algorithm works, what mathematical foundations it uses, and how it exploits both symmetry and asymmetry. We'll discover the remarkable contributions that Euler left for us.

Note:

1. This is where deep symmetries in number theory meet computational asymmetry. How fascinating that is!
2. More efficient cryptographic algorithms than RSA exist. Even though they may use different approaches for creating computational challenges, the core concept remains very similar. So examining the most intuitive(?) RSA algorithm as a case study makes perfect sense.
3. Having a daughter at a later age has brought me great joy, but watching her at just 3 years old already wanting to know everything about the world, I realize there may not be much time when we can have these kinds of conversations together. So I'm writing this hoping that someday, even if it's much later, she might be interested to see that her dad enjoyed these kinds of topics. Of course, right now my daughter would probably prefer that I just play Lego with her instead...