0

2

4

5

8

7

8

6

4

3

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

8

6

4

3

0

1

0

2

4

5

8

7

5

4

2

8

1

8

3

4

6

0

7

1

8

0

2

4

5

8

7

8

6

4

3

0

1

3

4

6

0

7

0

5

4

2

8

0

7

0

8

6

4

3

0

1

0

2

4

5

8

7

5

4

2

8

1

8

3

4

6

2

8

1

8

0

2

4

5

8

7

8

6

4

3

0

1

3

4

6

0

7

0

5

4

4

6

0

7

0

8

6

4

3

0

1

0

2

4

5

8

7

5

4

2

8

1

8

3

5

4

2

8

1

8

0

2

4

5

8

7

8

6

4

3

0

1

3

4

6

0

7

0

8

3

4

6

0

7

0

8

6

4

3

0

1

0

2

4

5

8

7

5

4

2

8

1

7

0

5

4

2

8

1

8

0

2

4

5

8

7

8

6

4

3

0

1

3

4

6

0

8

1

8

3

4

6

0

7

0

8

6

4

3

0

1

0

2

4

5

8

7

5

4

2

6

0

7

0

5

4

2

8

1

8

0

2

4

5

8

7

8

6

4

3

0

1

3

4

4

2

8

1

8

3

4

6

0

7

0

8

6

4

3

0

1

0

2

4

5

8

7

5

3

4

6

0

7

0

5

4

2

8

1

8

0

2

4

5

8

7

8

6

4

3

0

1

7

5

4

2

8

1

8

3

4

6

0

7

0

8

6

4

3

0

1

0

2

4

5

8

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

2

4

5

8

7

8

6

4

3

5

8

7

5

4

2

8

1

8

3

4

6

0

7

0

8

6

4

3

0

1

0

2

4

4

3

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

2

4

5

8

7

8

6

2

4

5

8

7

5

4

2

8

1

8

3

4

6

0

7

0

8

6

4

3

0

1

0

8

6

4

3

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

2

4

5

8

7

1

0

2

4

5

8

7

5

4

2

8

1

8

3

4

6

0

7

0

8

6

4

3

0

8

7

8

6

4

3

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

2

4

5

3

0

1

0

2

4

5

8

7

5

4

2

8

1

8

3

4

6

0

7

0

8

6

4

4

5

8

7

8

6

4

3

0

1

3

4

6

0

7

0

5

4

2

8

1

8

0

2

6

4

3

0

1

0

2

4

5

8

7

5

4

2

8

1

8

3

4

6

0

7

0

8

ANALYSIS
of the
REUT · REUT TRANSMAGIC SQUARE

by J. Frederic Teubner

The REUT · REUT (roy-tah · ro-toy) 12x12 Transmagic Square, in addition to having many amazing  properties as a pan-magic, pan-diagonal Magic Square (all rows, columns, diagonals, and broken diagonals sum to 60) is simultaneously a pan-magic, pan-diagonal Magic Table, all rows, columns, diagonals, and broken diagonals are evenly divisible by 3, 7, 13, and 91 in any of the eight compass directions, both forward and backwards, flipped and/or scrolled starting from any point on the chart (twelve cells required).

The REUT · REUT Transmagic Square can form a cube having continuous Magical and Transmagical properties on all faces both inside and out. The Transmagic Cube may then be unfolded and reversed to form an inverted cube having identical properties. The REUT · REUT can form of itself a new Transmagic Numberline and generate a new Square having identical properties. The Transmagic number may be extended by the coupling or intersection of one or more additional Transmagic numbers to create Transmagic Squares of ever increasing size.


THE MAGIC SQUARE:

On the REUT · REUT 12x12 Magic Square, the sum of each Row, each Column, and each of the two Major Diagonals is Sixty.


THE PAN-DIAGONAL SQUARE:

The Magic Sum of each of the broken diagonals (twelve cells to complete) is also Sixty.


THE PAN-MAGIC SQUARE:

Transfer a column(s) from side-to-side, or a row(s) from top-to-bottom and vice-versa. The reconfigured Square retains all  the magical and transmagical properties of the original.


SUM OF THREE BOXES:

The Magic Sum of Sixty may be obtained by adding together the digits contained within any three alternating boxes of four. In the top, left-hand corner of the REUT · REUT Square, highlight the very first four-square box - 9831 - skip the next box to 4346 - skip a box to 2488

Exception: on the left hand diagonal flow the three boxes are offset rather than being in the usual straight line arrangement.

THE FLOW OF NUMBERS:

There are three types of number-flow on the REUT · REUT;  the "Long" numbers, the "Tens," and the "Fifteens." The Long numbers occupy the columns and the rows, while the Tens (pairs adding to ten) and Fifteens (triplets adding to fifteen) flow along the opposing diagonals.

THE SUM OF THE DIGITS:

For every Long number (twelve cells) in every column and every row the sum of the digits is Sixty.

The Tens flow diagonally from the Left and require six pairs to complete a Magic string of Sixty. The same is true for every  broken Left diagonal.  It follows then that any six pairs of Tens may be combined in any pattern, order, or configuration to obtain the Magic sum.

The Fifteens flow along the opposing diagonal and require four sets of triplets to complete a Magic string of Sixty. The same is true for every broken Right diagonal. It follows then that any four sets of Fifteens may be combined in any pattern, order, or configuration to obtain the Magic sum.

The Tens, and the Fifteens may also be variously combined to produce the Magic sum of Sixty. Again, any workable pattern may be regarded as a template movable to any chart position.

TRANSMAGIC NUMBERS:

Every row, column, diagonal, or completed broken diagonal on the Reut.Reut is a twelve-digit Transmagic number.

A Transmagic number is a number with a specific factor or set of factors that are retained as the number is in various ways manipulated.  With few exceptions a Transmagic number may be flipped and/or scrolled to any position and then be evenly divided both forwards and backwards by a specific set of prime factors.

Example: 123123  Like ALL three-digit repeaters, this number 123-123 is evenly factored by 7, 11,  and 13 forwards and backwards.  It may be flipped: 321321, or scrolled: 231231 but it remains a three-digit repeater subject to the rule and Transmagic for 7, 11, and 13.

Example: 121212  Like ALL two-digit three-peaters, this number 12-12-12 is evenly factored by 7, 13, and 37  forwards and backwards.  It may be flipped: 212121, or scrolled: 212121 but it remains a two-digit three-peater subject to the rule and Transmagic for 7, 13, and 37.

THE TRANSMAGIC TABLE:

The Transmagic Table is a number matrix that is magic for Division only. There is no Magic Sum, only Magic Factors.

THE PAN-MAGIC TABLE:

Transfer a column(s) from side-to-side, or a row(s) from top-to-bottom and vice-versa. The reconfigured Table retains ALL the Transmagical properties of the original.

A Transmagic Table is evenly divisible by a Magic Factor or set of Magic Factors in any of the eight compass directions, both forwards and backwards, from any point on the chart (twelve cell wrap). The roster of Magic Factors varies by design from Table to Table.

THE TRANSMAGIC SQUARE combines the properties of a Magic Square with the properties of a Transmagic Table.

THE PAN-DIAGONAL TRANSMAGIC NUMBERS

Broken Diagonals: When reading a broken diagonal, one portion of the twelve-digit number is located on one side of the Major Diagonal and the second portion is located on the other side of the Major Diagonal.  Read the second portion in the same text direction as the first.  If reading the first portion from left-to-right then jump the Major Diagonal and read the second portion also from left-to-right to complete the twelve-digit transmagic number.

Example: If the first broken diagonal is eight cells in length,  jump the Major Diagonal to continue on the broken diagonal that is four cells in length, thus completing twelve cells.

THE LONG NUMBERS, as stated, are evenly divisible by 3, 7, 11, 13, 37, and 91 both forwards and backwards from any point on the numberline and may be scrolled to any position.  Example:

THE TENS, in sets of three like pairs such as 919191, are evenly divisible both forwards and backwards from any point on the numberline by 3, 7, 13, 37, and 91. Units may be repeated or linked such as 919191373737.

THE FIFTEENS, in sets of two like triplets such as 762762, are divisible both forwards and backwards from any point on the numberline by 3, 7, 11, 13, and 91. Units may be linked such as 762762186186.

FACTORS IN COMMON:

A quick comparison reveals that the LONG NUMBERS are evenly divisible by all six Magic Factors, but that the TENS are NOT evenly divisible by Magic 11 and the FIFTEENS cannot be evenly factored by Magic 37. Thus the Magic Factors common to all three flows are 3, 7, 13, and  91.

MOVABLE TEMPLATES:

ELEVENS

A. In the top, left-hand corner of the REUT · REUT Square, highlight the step-figure 9-8-1-2 read as 9812, 8129, 2981, 1298, or the reverse thereof. Each four-digit number formed is evenly factored by 11.  Move template to any position on the Square, read in similar fashion.

B. In the top, left-hand corner of the REUT · REUT Square, highlight the step-figure 9-3-1-7 read as 9317, 3179, 1793, 7931, or the reverse thereof. Each number formed is evenly factored by 11. Move template to any chart position.

THIRTY SEVENS

A. In the top, left-hand corner of the REUT · REUT Square, highlight the step-figure 9-8-8-1-2-2 read as 988122, 881229, 812298, 122988, or the reverse thereof. Each number formed is evenly factored by 37. Move template to any chart position.

B. In the top, left-hand corner of the REUT · REUT Square, highlight the step-figure 9-3-6-1-7-4 read as 936174, 361749, 617493, 174936, or the reverse thereof. Each number formed is evenly factored by 37. Move template to any chart position.

TRANSMAGIC LONG:

A. In the top, left-hand corner of the REUT · REUT Square, highlight the opening three numbers of the first ROW, 988. Next, highlight the three numbers on the third COLUMN directly beneath the 8, 2-9-3. On the fourth ROW , immediately following the 3, highlight the three numbers 1-2-2, below the final 2, highlight the three numbers 8-1-7. The step-figure 988293122817 is a Transmagic Long number

B. Similarly, in the top row of the REUT · REUT Square, highlight the three numbers 4-3-1, then highlight the top three numbers in the eighth column , 2-9-3. Next highlight the 6-7-9, fourth row. Lastly, highlight the numbers 8-1-7 in the eleventh column. The step-figure 431293679817 is a Transmagic Long Number.

C. At the top of the first column highlight the three numbers 9-3-6. Next, on the fourth row . highlight the numbers 6-4-3. On the fourth column highlight the numbers 1-7-4. Lastly, on the seventh row, highlight 4-6-7. The step-figure 936643174467 is Transmagic.

D. Similarly, At the top of the sixth column highlight the three numbers 3-6-6. Next, on the third row, highlight the numbers 4-3-1. O n the ninth column highlight the numbers 7-4-4. Lastly, on the sixth row, highlight 6-7-9. The step-figure 366431744679 is Transmagic.  Move templates to any chart position, reverse or transpose.

GENERATING A NEW TRANSMAGIC SQUARE:

Every Transmagic number formed by template can generate a new Transmagic Table (Magic Factors) but not necessarily a Transmagic Square (Magic Factors and Magic Sums.)  To generate a new Transmagic Square select any twelve-digit number from any column, row, or diagonal on the existing chart.

1. Form a 12 x 12 matrix

2. Enter the new twelve digit Transmagic number in the top row.  e.g. 

3. Flow in the Tens on the Left Diagonal the next row down e.g.    312246798864

4. Pick up the "6"        312246798864
                                   679886431224
5. Continue....                                                                                                                                                                                        


TEMPLATE SUMS:

Elevens: sum of the digits = 20 (one-third magic)

Thirty sevens: sum of the digits = 30 (one-half magic)

Transmagic Long: sum of the digits = 60 (magic)

Template sums adjust to the values of the Magic Sum for each chart, e.g. If the Magic Sum is 96, one third Magic=32 etc.  Like templates may be combined to form larger, more complex templates.

Template sums adjust to the Scale of each chart,  e.g. If the chart is 24x24 Tableau-1, the Template value for one- third Magic changes to become one-sixth Magic.  The Template for one-half magic changes to one-fourth Magic.

MATHEMAGIC:

Transmagic Squares contain layer upon layer of  mathemagical complexity.  There are no limits to their scale, variety, or digit-length, and they are infinite in number.  Creating Transmagic Squares that have the most possible common factors in each column, row, diagonal, or broken-diagonal while maintaining the magic sum is the pursuit of happiness.

Calc-u-lator Alligator!