The beautiful soaisu partition

Kenichi Takemura

19 April 2026

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Take a look at the array of numbers below. Can you tell what it is?

What a throwback

Yep, it’s the good old times table we all know and love.

Turns out, the times table we think we know inside out hides layer upon layer of astonishing structure that almost nobody has noticed. In this article I’m delighted to share just one of them with you. Right away, let’s focus on this blue area. What happens if we multiply every number in it? \[ 5 \cdot 6 \cdot 8 \cdot 14 \cdot 9 \cdot 24 \cdot 8 \cdot 36 \cdot 5 \cdot 50 \cdot 6 \cdot 60 \cdot 14 \cdot 63 \cdot 24 \cdot 64 \cdot 36 \cdot 63 \cdot 50 \cdot 60 \] Doing the calculation gives \[ 173401213127727513600000000, \] which doesn’t look obviously pretty. But if we rewrite it like this: \[ 173401213127727513600000000 = (10!)^4, \] the product is $10!$ raised to the fourth power. That’s far too suggestive to be a coincidence.

For the times tables, they are-a changin’

The key lies in a deeper property of the times table: pick one entry from each row and each column (ten numbers in total) and multiply them. The result is always $(10!)^2$:

\[ 1 \cdot 8 \cdot 9 \cdot 20 \cdot 50 \cdot 54 \cdot 42 \cdot 64 \cdot 63 \cdot 20 = 13168189440000 = (10!)^2. \]

This explains the diamond’s $(10!)^4$. See why? The diamond is not just a random shape; it is actually formed by combining two distinct sets of selections. In each set, we pick exactly one number from every row and every column.

Since the product of any such single selection is always $(10!)^2$, the product of the entire diamond region is simply \[(10!)^2 \times (10!)^2 = (10!)^4.\]

An entry in the $i$th row and $j$th column is simply the product $ij$. Selecting ten numbers such that each row and each column is represented exactly once is mathematically equivalent to choosing a permutation $\sigma$ of the set $\{1, 2, \dots, 10\}$. The product $P$ of these ten selected numbers can be written as \[ P = \prod_{i=1}^{10} (i \, \sigma(i)). \] Since $\sigma$ is a permutation (a reordering), the set $\{\sigma(1), \sigma(2), \dots, \sigma(10)\}$ is identical to the set $\{1, 2, \dots, 10\}$, just in a different order. Therefore, we can rearrange the product as \[ P = \left( \prod_{i=1}^{10} i \right) \left( \prod_{i=1}^{10} \sigma(i) \right) = 10! \times 10! = (10!)^2. \] The rows provide the `components’ $1$ through $10$, and the columns provide another set of $1$ through $10$. No matter how you pick the numbers, you are always multiplying the same set of parts. This is why the result is always $(10!)^2$.

We can pick cells from distinct rows and columns in multiple ways.

Using this fact, we can split the times table into five regions where the product of the numbers in each region is identical. The colouring above corresponds to the table below. Every colour group multiplies to the same value, \[(10!)^4 = 173401213127727513600000000.\]

And even better, the sums of the numbers in each set are also identical—each totals 605.

Isn’t it amazing that the times table can be split equally both by multiplication and addition using the same pattern? And the pattern stays unchanged under multiple $90^{°}$ rotations, left–right flips and up–down flips.

Groovy baby

The humble number grid

But the story doesn’t end here. This is only the warm-up… The real thrill comes next. This highly symmetric structure has far more power than we might imagine. Let’s look at something even more familiar: the $10 \times 10$ number grid, where the numbers $1$ to $n^2$ are simply filled in order.

Just the numbers 1 to 100 in order. Looks plain, right? Don’t be fooled. This grid hides breathtaking beauty. To reveal it, we need a new idea: soaisu.

The products of each set in the times table are all equal to $(𝟣𝟢!)^𝟦$ … and the sums of each set are all equal to 605.

Soaisu partitions

Soaisu are special partitions of distinct numbers where the sums of powers from 1 up to $k$ are equal across the sets. Formally, sets $S_1, S_2, \dots, S_m$ (each with $n$ integers) form a strength $k$ $\underbrace{n\text{–}n\text{–}\cdots\text{–}n}_{\text{$m$ times}}$ soaisu if for every $l = 1, 2, \dots, k$,
\[ \sum_{a \in S_1} a^l = \sum_{a \in S_2} a^l = \cdots = \sum_{a \in S_m} a^l. \]

Soaisu extends the classic Prouhet–Tarry–Escott problem (equal power sums for two sets) to three or more sets. I coined the term because I want every maths lover to know about them. The strength $k$ is affectionately written with $k$ hearts, ♥.

Seeing is believing, so here’s an example. Inside a $4 \times 4$ number grid lives a strength 3 8–8 soaisu ♥♥♥:

This particular soaisu has strength 3 since we can see that the sums of the first, second and third powers remain the same for both sets $A$ and $B$ but not for the fourth powers:

Yes—the numbers 1 to 16 split perfectly into two sets with equal power sums up to the cube. Every number lover should know this.

Number grids are soaisu squares

Back to the $10 \times 10$ number grid. Astonishingly, the numbers 1 to 100 can be split into five groups where the sums of the first, second, and third powers are all equal—using the exact same partition pattern from the times table! Let us call this arrangement type I.

Here are the groups:

\begin{align*} A = \{& 1, 10, 15, 16, 24, 27, 33, 38, 42, 49, 52, 59, 63, 68, 74, 77, 85, 86, 91, 100\}, \\ B = \{& 2, 9, 11, 20, 25, 26, 34, 37, 43, 48, 53, 58, 64, 67, 75, 76, 81, 90, 92, 99\}, \\ C = \{& 3, 8, 12, 19, 21, 30, 35, 36, 44, 47, 54, 57, 65, 66, 71, 80, 82, 89, 93, 98\}, \\ D = \{& 4, 7, 13, 18, 22, 29, 31, 40, 45, 46, 55, 56, 61, 70, 72, 79, 83, 88, 94, 97\}, \\ E = \{& 5, 6, 14, 17, 23, 28, 32, 39, 41, 50, 51, 60, 62, 69, 73, 78, 84, 87, 95, 96\}, \end{align*}

and here are the sums of the powers:

When expressed in the language of soaisu, this fact is represented as a 20–20–20–20–20 soaisu ♥♥♥. This notation concisely captures the essence of the structure: five groups, each consisting of 20 elements, are firmly bound together by a `soai power’ of strength 3.

The number grid is coloured in accordingly:

The natural numbers in a grid…

… now with added colour

To think that the numbers 1 to 100 could be partitioned equally into five sets as soaisu ♥♥♥!

When I first discovered this, I could hardly believe my eyes. I was certain that such a miraculous partition must be unique. However, as is often the case in mathematics, my intuition was about to be betrayed.

Now there are two of them!

Incredibly, another version exists: type II, a pattern that appears to be the dual of type I! While the long journey to this discovery is a story for another time, this second miraculous arrangement can be obtained by tiling four copies of the original pattern together.

Four tiles of type I reveal type II.

Take a look at the tiled pattern above. The bold black border marks the type II pattern. Just like type I, this layout is invariant under rotations of $90n^{°}$ (where $n$ is an integer) and reflections.

When we apply this pattern to the $10 \times 10$ number grid, we get the colouring below.

X marks the spot: we can analyse the sets in type II as well.

The groups are

\begin{align*} A = \{&1, 10, 12, 19, 23, 28, 34, 37, 45, 46,\\ & 55, 56, 64, 67, 73, 78, 82, 89, 91, 100\}, \\ B = \{&2, 9, 13, 18, 24, 27, 35, 36, 41, 50,\\ & 51, 60, 65, 66, 74, 77, 83, 88, 92, 99\}, \\ C = \{&3, 8, 14, 17, 25, 26, 31, 40, 42, 49,\\ & 52, 59, 61, 70, 75, 76, 84, 87, 93, 98\}, \\ D = \{&4, 7, 15, 16, 21, 30, 32, 39, 43, 48,\\ & 53, 58, 62, 69, 71, 80, 85, 86, 94, 97\}, \\ E = \{&5, 6, 11, 20, 22, 29, 33, 38, 44, 47,\\ & 54, 57, 63, 68, 72, 79, 81, 90, 95, 96\}. \end{align*}

The sums of the first to third powers remain `pinned’ to the same values, as if by magic. They are:

Who would have thought such elegant symmetry — doubled, no less — lurked behind these numbers? This is pure design crafted by the numbers themselves. One reason we find symmetry beautiful is right here.

Perhaps the robust symmetry embedded within these patterns is the very key to unlocking the secrets of soaisu. Indeed, a group structure known as the dihedral group $D_4$ lies latent here.

A thrilling thought flashed through my mind: `Using this idea, might it be possible to achieve equal partitions of soaisu in grids of sizes other than $10 \times 10$?’ Many of you may have already noticed that the 8–8 soaisu ♥♥♥ pattern also possesses the same kind of symmetry.

I have been travelling through the world of soaisu for over 15 years, and the more I learn, the more profound I find their structure to be. Beyond the dihedral group, cyclic groups and symmetric groups pull the strings behind the scenes in many different ways. Moreover, soaisu find their home not only in number grids like these, but also in magic squares.

Oh dear, I could go on forever. This is becoming a long story, but let’s save that for another time. I hope to see you again soon!

Kenichi Takemura

Kenichi is is an independent researcher based in Tokyo, Japan, specialising in soaisu and magic squares. When he is not uncovering hidden symmetries in numbers, he is likely out for a walk. Rain or shine, he aims to walk exactly 11,111 steps every day for his health.

All articles by Kenichi