Mathematics is not about absolute, formulaic, only-one-way methods. It is about exploration.

We often forget to emphasize the creativity flowing through math, the skills of reading and understanding a question before setting out to solve it, and the continuous wondering and wandering involved as we solve it. We talk about these essential steps, but rarely model them. Or we model them occasionally in hopes our students know they should be applying these steps with every lesson, even if we don’t.

In the last two posts, we explored the determination of and connection between several distinct answers to an All-digits Arithmetic challenge posed in 2011 by Dennis Coble, a fellow tweeter and friend of mine.

Over the years several people have contributed to the solving of Coble’s challenge, including in the beginning, Ross Mannell, Earl Samuelson and Dennis himself. Others have further contributed to or used the problem, which led to me create the math lab associated with this challenge.

One of the more recent explorers and contributors has been Joe Lubich, a Google Plus and MeWe friend who I met in a Genetics and Evolution MOOC offered by Mohamed Noor.

Joe was the person who grouped the distinct answers into the nations and stabiles described at the end of the last post. He also investigated the extension of the challenge we will explore in this post.

*Update: Actually, the extension will be explored in my next post. Here we will quickly explore how Joe worked through the All-digits Arithmetic challenge and the patterns he uncovered. This exploration is distinct enough from the extension Joe explored to warrant its own post. It also shortens the post into manageable pieces.*

The following patterns make more sense after you read the first two posts analyzing this challenge.

Analyzing viable totals

Joe began solving the All-digits Arithmetic challenge by seeking answers that included two three-digit addends and a three-digit total. He found scattered answers, but he never knew how these answers related and whether more existed. So, a direct full-solution seems untenable.

When addition carried him only so far, he concentrated on finding viable totals (minuends for those exploring this challenge through subtraction).

He began by listing all the three-digit numbers from 000 to 999 in a 10 x 100 grid. He then circled the numbers with digital sums of 18; he actually hadn’t figured out that all correct totals had digital sums of 18 yet, but he noticed a pattern of correct answers (full additions) that highlighted these.

In his own words,

In an attempt at a new approach I listed all my [totals] in Excel and then sorted them from smallest to greatest. Then I found the difference between each [total] and the previous [total]. I was surprised to see that every difference was a multiple of 9, and then I noticed that every [total] was also a multiple of 9.

The three-digit numbers with digital sums of 18 line up in diagonals in nine-diagonal intervals.

Since all other numbers were distractions, Joe collapsed his sieve to include only those numbers he highlighted.

I skewed the resulting triangle to create this grid.

Notice, the original diagonals still line up in positively pitched inclinations.

To emphasize patterns, in this grid, I

- accented those totals containing the same digits with the same coloured background,
- included numbers, marked in red, which fit the grid, but are not viable totals,
- bracketed numbers with repeating digits and gave them grey fonts and white backgrounds, and
- (in the grids below) dropped 099, 909 and 990, since these had zero in them (and repeating digits).

**Pattern: Viable-total symmetry**

Glancing at the grid, you will notice a three-axis symmetry to it that divides the grid into six congruent, albeit chiral, parts. The axes of symmetry run through the numbers containing repeating digits (white backgrounds).

Notice the multi-way symmetry of the numbers in these six parts. Not only are the shapes of the parts congruent, so are the positioning of numbers with common digits.

This is emphasized in the following rendition of the grid.

In the grid above, the numbers are replaced by letters corresponding to the numbers’ background colours: L = blue, P = purple, R = red, Y = yellow, O = orange, B = brown and G = green. The mirror-symmetry of the grid parts and position-symmetry of numbers with common digits are much clearer. We not only see the radiant symmetries in this grid, the letters ripple out from the centre: L (blue 5,6,7) first to G (green 1,8,9) last.

The concentric arrangement of the numbers is a by-product of their radiant symmetry. A systematic repeat of numbers containing the same digits (L P R Y, L P R instead of L P R Y O B G) would warrant a deeper analysis. This is particularly true if the pattern were something like L P R Y R P L, a concentric symmetry. In both cases, exploring why the totals fall in this pattern and how the totals consequently relate would be fascinating.

Take a moment to appreciate the complex symmetries in this grid. Even though they suit the nature of the numbers, they continue to amaze me artistically and reveal depths I had not appreciated to the relationships of the viable totals.

The exchange of digits follows a pattern too, owing to the periodicity of the numbers, and reveals yet another symmetry. In each row, the first digit remains constant while the other two switch. In each positive diagonal, the mid digit stays the same. And in the negative diagonals, the last digit does not change. Further, the constant digits climb from 1 to 9 vertically and descend from 1 to 9 down each diagonal (corners are missing because the numbers that would occupy them — 909, 990 and 099 — have repeating nines and zero in them).

This produces a symmetry along the positive diagonals, where the mid digit is retained and the first and last digits switch. The first and last numbers of each diagonal reverse each other. So too the second and second last numbers, the third and third last, and so on to the middle of the diagonal, owing again to the constant periodicity of the numbers in the grid.

**Pattern: Distinct-total symmetry**

Lubich’s sieve contains 31 viable totals that resolve Coble’s challenge. Not all these sums are distinct as defined in my last post. Some of them are those variants where the free column, that neither lends nor borrows, is in the units place instead of the hundreds one.

This is the distribution of the distinct totals.

Notice here that the distinct totals tend toward the numbers, of greater value, to the upper right of the grid and are symmetrically grouped into nations.

- The
**a**,**b**,**c**,**l**,**g**,**h**,**i**and**d**,**e**,**f**nations border the positive-diagonal axis of symmetry with each of**a**,**c**,**b**,**l**,**g**,**i**and**d**,**f**sharing the same totals mostly below the axis. **u**,**t**,**n**,**m**and**k**,**j**,**q**,**s**form three nations that hug the top and inclined-right edges of the grid. The answers associated with the first three pairs of totals are related by horizontal exchanges. The last,**q**,**s**, is rotationally related.**o**,**p**and**r**also share common digits. They are set in from the grid edges, rotational**p**and**r**beyond**q**and**s**and the main mass of distinct totals,**o**seemingly out of place, but symmetric with 684, which is not a viable total.

The totals with the floating free-digit (that neither lends nor borrows) in the units place have a similar distribution.

They tend toward the upper left of the grid, with “middle” number values, and are symmetrically flipped from the distinct totals.

No viable totals are found in the bottom centre of the grid, which contains numbers of lesser value. Also, as illustrated in the grid below, some distinct totals and free-units totals share common viable values.

It turns out the grouping of nations (via answer totals) is as interesting as the grouping of petals (via answer relationships). Since Joe followed his own solution to solving Coble’s challenge, the patterns unveiled by the nations illustrate more relationships among the totals of correct answers to the challenge than those explored in the last post.

It is possible to discover and explore these novel approaches and novel relationships because the All-digits Arithmetic challenge is an open exploration and not just another math exercise.

**Which approach to solving Coble’s challenge did you find most enlightening? Please, comment and ask questions below. I look forward to seeing what you have to ask and say.**