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Reverse Divisors and Magic Numbers

This paper explores the intriguing mathematical relationship between Ball's magic numbers and reverse divisors, highlighting how these properties can be used to create engaging and surprising activities for mathematics students.

Original authors: Eudes Antonio Costa, Ronaldo Antônio Santos

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eudes Antonio Costa, Ronaldo Antônio Santos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where numbers aren't just cold, hard facts, but characters in a playful story with hidden personalities. This paper, written by Costa and Santos, is like a detective story that uncovers the secret relationships between two very specific groups of numbers: "Magic Numbers" and "Reverse Divisors."

Here is the story of their discovery, broken down into simple concepts.

1. The Magic Trick: "Ball's Magic Numbers"

First, the authors introduce a fun party trick called Ball's Magic Numbers. Imagine you have a number, say 572.

Flip it: Write it backwards to get 275.
Subtract: Take the bigger one and subtract the smaller one ( $572 - 275 = 297$ ).
Flip the result: Turn 297 into 792.
Add them up: $297 + 792 = 1089$ .

No matter what three-digit number you start with (as long as it's not a palindrome like 121), if you follow these steps, you will always end up with 1089. If you start with two digits, you always end up with 99.

The authors call these final results "Magic Numbers." They are like the "grand prize" of a math game. The paper reveals that these numbers are actually just multiples of 99 (like 99, 1089, 9999, etc.).

2. The Secret Code: "Zeros and Ones"

How do we know exactly which numbers are Magic Numbers? The authors found a secret code.

Think of the subtraction step in the magic trick as a game of "borrowing." When you subtract digits, sometimes you have to borrow a 10 from the neighbor. The authors realized that if you write down a 1 every time you borrow and a 0 when you don't, you create a secret string of zeros and ones (like 11010).

This string is the "Code."

If you take this code and multiply it by 99, you get a Magic Number.
It turns out these codes are very picky. They can only be made of 0s and 1s, and they have to follow strict rules about where the 1s and 0s can sit. It's like a password that only certain numbers know.

3. The Mirror Image: "Reverse Divisors"

Next, the paper introduces Reverse Divisors.
Imagine a number that is a "mirror" of another. If you take a number, flip it backwards, and the new number is perfectly divisible by the original, they are a special pair.

Example: Take 1089. Flip it to get 9801.
Does 1089 go into 9801 evenly? Yes! $9801 \div 1089 = 9$ .
So, 1089 is a "Reverse Divisor."

The paper proves that these numbers are rare. You won't find any with just 2 or 3 digits. They usually start appearing at 4 digits (like 1089, 2178) and then follow a pattern where you just stuff a bunch of 9s in the middle (like 1099989).

4. The Big Reveal: They Are the Same Family!

This is the "Aha!" moment of the paper. The authors discovered that Magic Numbers and Reverse Divisors are actually the same family.

The number 1089 is a Magic Number (it's the result of the subtraction game).
The number 1089 is also a Reverse Divisor (because its reverse, 9801, is a multiple of it).

The paper shows that many Magic Numbers are actually Reverse Divisors. They are like twins separated at birth who look different but share the same DNA (the secret code of 0s and 1s).

5. Building Blocks: Repunits and Undulating Numbers

To explain how these numbers are built, the authors use two fun concepts:

Repunits: Numbers made entirely of 1s (like 1, 11, 111, 1111).
Undulating Numbers: Numbers that wave up and down, alternating between 1 and 0 (like 101, 10101).

The paper shows that you can build these magical Reverse Divisors by multiplying the basic Magic Number (1089) by these "waving" numbers. It's like taking a Lego brick (1089) and snapping different colored blocks (101, 10101) onto it to create new, larger structures.

6. The Square Number Mystery

Finally, the authors ask a question: "If I multiply two different Reverse Divisors together, do I get a perfect square number (like 4, 9, 16, 25)?"

They run the numbers and say No.
It's like trying to build a perfect square tile out of two different, irregular puzzle pieces. No matter how you combine two different Reverse Divisors, you will never get a perfect square. The only time you get a square is if you multiply a number by its own reverse (e.g., $1089 \times 9801$ ), but that's a different story.

Summary

In simple terms, this paper is a celebration of number patterns. It shows that:

There is a fun game that always ends with the number 1089 (or 99).
There is a special group of numbers that are divisible by their own mirror images.
These two groups are actually the same thing.
They are built using secret codes of 0s and 1s, and they can be expanded by adding "waves" of 1s and 0s.

The authors conclude that these numbers are a great way to make math fun and surprising, proving that even in the rigid world of numbers, there is room for magic and curiosity.

Technical Summary: Reverse Divisors and Magic Numbers

Problem Statement
This paper investigates the mathematical relationship between two specific classes of integers: Ball's magic numbers and reverse divisors (also known as palintuples). While both classes are defined through operations involving the reversal of digits, they originate from different contexts. Ball's magic numbers arise from a specific algorithmic process involving the difference between a number and its reverse, followed by adding the reverse of that difference. Reverse divisors are integers that divide their own digit-reversed counterparts. The study aims to characterize these numbers, establish their properties, and identify the specific intersection where a number belongs to both classes.

Methodology
The authors employ a combination of algorithmic definition, number-theoretic analysis, and combinatorial characterization using binary codes.

Algorithmic Definition: The paper formalizes "Ball's magic numbers" via Algorithm 1. For a non-palindromic $n$ -digit number $x_n$ , the algorithm calculates $y_n = |x_n - x'_n|$ (where $x'_n$ is the reverse) and then computes $B = y_n + y'_n$ .
Code Characterization: Following the work of Webster [2], the authors introduce a "code" system ( $Z_n$ ) consisting of sequences of 0s and 1s. This code is derived from the borrowing operations required when subtracting a number from its reverse. The paper establishes that every Ball's magic number $B$ can be expressed as $B = 99 \times Z_{truncated}$ , where $Z_{truncated}$ is the binary code with the leading 1 and trailing 0 removed.
Reverse Divisor Analysis: The study defines reverse divisors as non-palindromic integers $x_n$ such that $x'_n = k \cdot x_n$ for some integer $1 < k < 10$ . The authors utilize induction and algebraic manipulation to prove the existence and structure of these numbers, specifically focusing on forms involving repunits (numbers consisting only of the digit 1).
Intersection Analysis: The paper analyzes the conditions under which a number satisfies the definitions of both a Ball's magic number and a reverse divisor.
Square Number Investigation: The final section investigates whether the product of two distinct reverse divisors can result in a perfect square, utilizing properties of repunits and greatest common divisors.

Key Contributions and Results

Characterization of Ball's Magic Numbers: The paper confirms that all non-zero Ball's magic numbers are multiples of 99. It provides a rigorous characterization linking these numbers to binary codes ( $Z_n$ ) composed of 0s and 1s. Specifically, it proves that the number of possible Ball's magic numbers for a given digit length relates to sums of Fibonacci numbers.
Structure of Reverse Divisors: The authors prove that there are no reverse divisors with 2 or 3 digits. They establish that for $n > 3$ $n > 3$ , reverse divisors exist in two primary forms:
- $11 \times (10^{n-2} - 1)$ , which results in numbers of the form $1099\dots989$ .
- $22 \times (10^{n-2} - 1)$ , which results in numbers of the form $2199\dots978$ .
  The paper provides inductive proofs (Theorems 12 and 13) demonstrating that these forms are indeed reverse divisors with quotients of 9 and 4, respectively.
Intersection of Classes: A central finding is that specific reverse divisors are also Ball's magic numbers. Theorem 20 demonstrates that numbers of the form $11 \times (10^n - 1)$ are both reverse divisors and Ball's magic numbers. Consequently, the paper shows that certain reverse divisors can be decomposed into the sum of two other magic numbers (Theorem 21).
Divisibility Properties: The study reveals that reverse divisors of the form $11 \times (10^{2n} - 1)$ possess at least $n+1$ magic numbers as divisors (Theorem 22).
Undulating Numbers: The paper connects "undulating numbers" (alternating 1s and 0s, e.g., 10101) to these classes, showing that products of 1089 and specific undulating numbers yield both magic numbers and reverse divisors.
Non-Square Product: The paper proves that the product of two distinct reverse divisors is never a perfect square (Theorem 26), relying on the fact that repunits $R_n$ for $n > 1$ are not perfect squares.

Significance and Claims
The authors frame the significance of this work within the context of recreational mathematics. They claim that the study reveals "beautiful and curious properties" and "surprising connections" between seemingly distinct numerical classes. The paper posits that these activities serve as a motivational tool for mathematics students, enabling the analysis of deep number-theoretic relationships through accessible algorithms.

The paper modestly claims to present the properties and connections between these classes, specifically identifying the subset of numbers that are simultaneously Ball's magic numbers and reverse divisors. It does not propose new applications outside of mathematical education and recreational analysis but asserts that the established connections (involving Fibonacci numbers, repunits, and binary codes) offer a structured way to understand these curious numerical phenomena. The authors conclude by suggesting that the "generosity" of mathematics implies further interesting properties involving these numbers remain to be discovered.