Number Puzzle TM
A Mathematical Learning System
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Substitutes for a a Substitutes for b b Substitutes for c c Substitutes for d d Tetractys abcd f 2 f 2 f 1 f 1

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Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 8 6 2. x = c d 2 4 Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 8 2 4. x = b d 6 4

How to Play (detailed instructions below)
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 (f1), 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 (f2), 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.

Instructions for playing the Tetractys Number Puzzle

Background information

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.)

 ×  0  1  2  3  4  5  6  7  8  9 0 00 00 00 00 00 00 00 00 00 00 1 00 01 02 03 04 05 06 07 08 09 2 00 02 04 06 08 10 12 14 16 18 3 00 03 06 09 12 15 18 21 24 27 4 00 04 08 12 16 20 24 28 32 36 5 00 05 10 15 20 25 30 35 40 45 6 00 06 12 18 24 30 36 42 48 54 7 00 07 14 21 28 35 42 49 56 63 8 00 08 16 24 32 40 48 56 64 72 9 00 09 18 27 36 45 54 63 72 81

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

Understanding the rule

This puzzle rules

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):

Substitutes for a Substitutes for b Substitutes for c Substitutes for d a b c d Tetractys 1 - 737 abcd f 2 f 2 f 1 f 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 (f1), 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 (f2), 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.

Analyzing the rule

A formal introduction is in order

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):

3 2 2 8 Column Form Row Form Column 1 Column 2 f 2 f 2 f 1 f 1 3 2 2 8 Row 1 Row 2 f 2 f 2 f 1 f 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):

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 3 2 2. x = c d 2 8

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 f1 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 f1 digits.

If we follow what the rule requires, we need to find all the f1 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 f1 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).

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 3 2 2. x = c d 2 8 4 4 1 0

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.

Let’s try a different angle

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.

Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 3 2 4. x = b d 2 8

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


In the row form of the rule, then, the f2 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).

4 4 4 4 3 2 2 8 Tetractys 164 3228 f 2 f 2 f 1 f 1

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


Notice that we repeat twice each f1 and f2 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.

How’s that again?

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.

8 6 2 4 Column Form Row Form Column 1 Column 2 f 2 f 2 f 1 f 1 8 6 2 4 Row 1 Row 2 f 2 f 2 f 1 f 1

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).

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 8 6 2. x = c d 2 4 Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 8 2 4. x = b d 6 4

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


When we search for the f1 digits to solve equations 1 and 2, we find that the f1 digits are 2 and 7. If we insert these digits back into our equations, we get the following result (Fig. 3-8):

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 8 6 2. x = c d 2 4 2 2 1 0 1. x = a b 8 6 2. x = c d 2 4 7 7 5 1

Fig 3-8: Solution of f1 digits in column form of Tetractys 573


When we search for the f2 digits to solve equations 3 and 4, we find that the f2 digits are 4 and 9. If we now insert these digits back into our equations, we get the following result (Fig. 3-9):

Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 8 2 4. x = b d 6 4 4 4 3 2 3. x = a c 8 2 4. x = b d 6 4 9 9 7 5

Fig 3-9: Solution of f2 digits in row form of Tetractys 573


Transferring the f1 and f2 digits back into the puzzle grid for Tetractys 573, we show the solution as follows (Fig. 3-10):

49 49 2 7 2 7 8 6 2 4 Tetractys 573 8624 f 2 f 2 f 1 f 1

Fig 3-10: Solutions of f1 and f2 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 f1 and f2 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.

Variety is the spice of life

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:

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 3 2 2. x = Substitutes for d c d 2 Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 3 2 4. x = b d 2

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 f1 digit and one f2 digit to make our solution work, since we don't need to find all f1 and f2 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).

Column Rule Form Column 2 Column 1 f 1 10s 1s/10s 1s 1. x = a b 3 2 2. x = c d 2 1 7 7 1 4 Row Rule Form Row 2 Row 1 f 2 10s 1s/10s 1s 3. x = a c 3 2 4. x = b d 2 1 7 7 1 4

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


When f1 and f2 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).

Substitutes for a a 38 Substitutes for b b 27 Substitutes for c c 27 18 4 4 4 4 Substitutes for d d 3 2 2 8 Tetractys 164 3228 f 2 f 2 f 1 f 1

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


Solving the online puzzle

A couple of conventions to keep in mind

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 f2 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!