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The puzzle’s simple rule states that all four digits (abcd) are related
to one another in such a way that both these rules are true:
Column Rule

When you multiply both digits of the first column (a and
c) by some number from 0 to 9 (f_{1}), the corresponding digits
of the second column (b and d) are from either the ones position
of both products or the tens position of both products.

Row Rule

When you multiply both digits of the first row (a and b)
by some number from 0 to 9 (f_{2}), the corresponding digits of
the second row (c and d) are from either the ones position
of both products or the tens position of both products.

The Tetractys™ Number Puzzle (pronounced ‘**tuh-TRACK-tiss**’)
gets its name from an ancient Greek word meaning ‘fourfold’
or ‘set of four’. The word ‘tetractys’ was famously used by the
followers of Pythagoras for the first four numbers—**1**, **2**, **3**, and
**4**—because these numbers were found in the musical ratios of the
octave (**2:1**), perfect fifth (**3:2**), and perfect fourth (**4:3**). But other
sets of four things have also been compiled around the word
‘tetractys’. The four seasons (spring, summer, autumn, winter)
form a ‘tetractys’, as do the four geometric dimensions (point,
line, plane, solid), the four Gospels (Matthew, Mark, Luke, John),
and the four elements of the ancient world (air, fire, earth, water).

The Tetractys Number Puzzle, however, is based on sets of four
digits, each from **0** to **9** and related to one another by a simple,
arbitrary rule. It is this rule which defines every puzzle and
makes each digit in a set of four significant. In fact, the rule of the
Tetractys Number Puzzle is so simple, that all that is required to
understand it is a basic knowledge of the multiplication table up
to **9 × 9** and the decimal place value system. Because this knowledge
is so important for understanding the puzzle, we will show
it below in tabular form (Fig. 1-1). (**Note**: All products in the table
show ones and tens position values, even if the tens value is **0**.)

**Fig 1-1**: Multiplication table of decimal digits from **0** to **9**

There are 737 different Tetractys puzzles in the Tetractys Number Puzzle. Each puzzle has four digits (abcd) from 0 to 9 arranged in two rows and two columns following a simple rule repeated twice: once for the columns and once for the rows. The puzzle grid basically looks like this (Fig. 2-1):

**Fig 2-1**: Basic grid form of the Tetractys Number Puzzle

The puzzle’s simple rule states that all four digits (abcd) are related to one another in such a way that:

Column Rule

When you multiply both digits of the first column (a and
c) by some number from 0 to 9 (f_{1}), the corresponding digits
of the second column (b and d) are from either the ones position
of both products or the tens position of both products.

Row Rule

When you multiply both digits of the first row (a and b)
by some number from 0 to 9 (f_{2}), the corresponding digits of
the second row (c and d) are from either the ones position
of both products or the tens position of both products.

As long as you know both the multiplication table up to 9 × 9 and
how the decimal place value system works, you can understand,
analyze, and solve any Tetractys puzzle. *It’s that simple!*

This rule is the heart of the Tetractys puzzle, because only after we understand it will we then be able to properly analyze the puzzle and then solve it.

Having understood the rule governing every Tetractys puzzle, our next task is to analyze the four digits of the puzzle so that we can solve it. Let’s look at an example puzzle to illustrate the kind of analysis we need to do. Here is Tetractys 164 broken down into both column and row forms (Fig. 3-1):

**Fig 3-1**: Column and row forms of the Tetractys puzzle grid

According to the rule of the Tetractys puzzle in column form, we must first look at the column digits and see how they relate to one another. We can analyze them into two simple multiplication equations as follows (Fig. 3-2):

**Fig 3-2**: Column form with multiplication equations

The first column digits (3 and 2) are shown in both equations in
the blue boxes, the second column digits (2 and 8) are shown
in the yellow boxes, and the unknown factors to solve for, the f_{1}
digits, are to be placed in the orange boxes. The two blank spaces
on each side of the second column digits represent the remaining tens
or ones position digits of the products produced by the first column
digits and the f_{1} digits.

If we follow what the rule requires, we need to find all the f_{1} digits
that, when multiplied by the first column digits (blue boxes), produce
the second column digits (yellow boxes) as either ones or tens position
digits. As it turns out in this example, there is only one such f_{1}
digit that meets this requirement and it is 4. Let’s now insert this digit
back into our two equations to see what result we get (Fig. 3-3).

**Fig 3-3**: One solution to the column form of the Tetractys rule

We can see why the digit 4 works in this example, because the two products, 12 and 8, have their ones position digits identical to those of the second column, 2 and 8. The digit 4, therefore, is the solution to the column form of the rule.

What about the row form of the rule? If we do the same analysis as we did for the column form (Fig. 3-4), we notice that our two multiplication equations are actually the same! This is because, if we exchange the columns for rows, the columns and rows will still be the same.

**Fig 3-4**: Row form of the Tetractys rule with multiplication equations

In the row form of the rule, then, the f_{2} digit that validly conforms
to the rule is also 4, since the equations in both column and row
form are the same. We can now show how this solution is shown in
the grid form of the Tetractys puzzle (Fig. 3-5).

**Fig 3-5**: Solutions to the column and row forms shown in orange cross

Notice that we repeat twice each f_{1} and f_{2} digit in the orange cross
pattern to show that each of these digits is multiplied twice, once
for each multiplicand in the blue boxes of our four equations.

Let’s now look at another Tetractys puzzle where the column and row forms are not identical. Fig. 3-6 shows Tetractys 573 as an example.

**Fig 3-6**: Non-identical column and row forms of Tetractys 573

Since we have already explained how to analyze these forms in our analysis of Tetractys 164, let’s simply show the four multiplication equations that we can derive from the column and row forms above for Tetractys 573 (Fig. 3-7).

**Fig 3-7**: Multiplication equations for column and row forms of Tetractys 573

When we search for the f_{1} digits to solve equations 1 and 2, we find
that the f_{1} digits are 2 and 7. If we insert these digits back into our
equations, we get the following result (Fig. 3-8):

**Fig 3-8**: Solution of f_{1} digits in column form of Tetractys 573

When we search for the f_{2} digits to solve equations 3 and 4, we find
that the f_{2} digits are 4 and 9. If we now insert these digits back into
our equations, we get the following result (Fig. 3-9):

**Fig 3-9**: Solution of f_{2} digits in row form of Tetractys 573

Transferring the f_{1} and f_{2} digits back into the puzzle grid for Tetractys
573, we show the solution as follows (Fig. 3-10):

**Fig 3-10**: Solutions of f_{1} and f_{2} digits displayed for Tetractys 573

At this point we have shown how to analyze the Tetractys rule in
any particular Tetractys puzzle, so that we can solve for any of the
f_{1} and f_{2} digits that validly conform to the Tetractys rule. For those
who want an additional challenge, however, we will have to show
another way of solving the Tetractys puzzle.

Another variation of the Tetractys Number Puzzle involves solving the puzzle by finding substitute digits for each of the four digits of the puzzle (abcd). For example, given Tetractys 164 above, which has the digits 3228, suppose we want to know if there are any other valid Tetractys puzzles that would satisfy the form 322d, where d can be any digit from 0 to 9. We already know that 8 will satisfy the rule of the Tetractys puzzle, but are there any others?

If we do an analysis using multiplication equations similar to the previous examples, we would end up solving something like the equations shown in Fig. 3-11:

**Fig 3-11**: Multiplication equations with blank spaces for substitutes of digit d.

These equations look a bit more tricky, because we have more colored
boxes to fill in and solve. However, we only need to find one
f_{1} digit and one f_{2} digit to make our solution work, since we don't
need to find all f_{1} and f_{2} digits. As long as we find one of each, we
will know that the digit we find for d (the empty yellow boxes)
makes 322d a valid Tetractys puzzle.

It turns out that in this example there is only one other digit other than 8 which will work, and it is the digit 1. Let's now show how this digit works in the equations we started out with (Fig. 3-12).

**Fig 3-12**: Multiplication equations with solution for substitutes of digit d.

When f_{1} and f_{2} digits are both 7 and d is 1, the equations are all
solved, so that the second column and second row digits represent
the tens position digits of the corresponding products, 21 and 14.

In fact, we can do the same analysis for each of the remaining digits of Tetractys 164—that is, for a228, 3b28, and 32c8. If we were to find all these other solutions, the final puzzle form would then display these solutions in each of the four corners of the puzzle grid (Fig. 3-13).

**Fig 3-13**: Solutions shown of substitute digits of Tetractys 164

One convention to make it easier to identify particular puzzles is the positioning of the four digits of each puzzle to the right side of the puzzle. This provides a convenient way of naming each puzzle, so that Tetractys 164, for example, can also be referred to as '3228'. Since each Tetractys puzzle has a unique reference number (from 1 to 737) and a unique set of four digits, this alternative reference name will also be unique.

Finally, one other convention has been used in this online puzzle in order
to reduce number clutter in those puzzles where there are many
solution digits that represent consecutive numbers. Let's suppose
that the f_{2} digits of a particular puzzle are the digits 1 through 9.
This would be shown as '123456789' in the horizontal orange strip
of the puzzle grid and would look rather crowded, especially since
these digits would be repeated twice. Instead of displaying all
these consecutive digits, the convention would show only the first
and last digits with a dot between them to represent all digits
between these two numbers. If the last digit is 9 or the first digit is
0, these first and last digits are also excluded. Thus we would see
'1•' instead of the longer '123456789'. Here are examples of how
this convention works throughout the book.

'01234' becomes '•4'

'012346789' becomes '•46•'

'4678' becomes '46•8'

'0123456789' becomes '•'

'012' becomes '•2', not '0•2'

But...

‘89’ remains ‘89’, not ‘8•’

‘024689’ remains ‘024689’

‘124578’ remains ‘124578’

All that's left to do now is to solve all 737 Tetractys puzzles in this online puzzle!