ANALYSIS
of the
REUT · REUT TRANSMAGIC SQUARE
by J. Frederic Teubner
The REUT
· REUT (roytah · rotoy) 12x12 Transmagic Square, in addition to
having many amazing properties as a panmagic, pandiagonal Magic
Square (all rows, columns, diagonals, and broken diagonals sum to 60)
is simultaneously a panmagic, pandiagonal 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 PANDIAGONAL SQUARE:
The Magic Sum of each of the broken diagonals (twelve cells to complete) is also Sixty.
THE
PANMAGIC SQUARE:
Transfer a column(s) from sidetoside, or a row(s) from toptobottom
and viceversa. 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,
lefthand corner of the REUT · REUT Square, highlight the very first
foursquare box  9831  skip the next box to 4346  skip a box to 2488
The
sum of the digits is Sixty.

_{9}

_{8}



_{4}

_{3}



_{2}

_{4}



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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 numberflow 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 twelvedigit 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 threedigit repeaters, this number 123123 is
evenly factored by 7, 11, and 13 forwards and backwards. It may be
flipped: 321321, or scrolled: 231231 but it remains a threedigit
repeater subject to the rule and Transmagic for 7, 11, and 13.
Example:
121212 Like ALL twodigit threepeaters, this number 121212 is
evenly factored by 7, 13, and 37 forwards and backwards. It may be
flipped: 212121, or scrolled: 212121 but it remains a twodigit
threepeater 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 PANMAGIC TABLE:
Transfer a column(s) from sidetoside, or a row(s) from toptobottom
and viceversa. 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 PANDIAGONAL TRANSMAGIC NUMBERS
Broken Diagonals: When
reading a broken diagonal, one portion of the twelvedigit 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 lefttoright then jump the Major Diagonal and
read the second portion also from lefttoright to complete the twelvedigit 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
opening line of The REUT · REUT is

_{9}

_{8}

_{8}

_{6}

_{4}

_{3}

_{1}

_{2}

_{2}

_{4}

_{6}

_{7}

Evenly
Divisible by 3, 7,11,13, 37, and 91.

Backwards

764221346889

Backwards
from "1"

134688976422

Forward
from "7"

798864312246

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, lefthand corner of the REUT · REUT Square, highlight
the stepfigure 9812 read as 9812, 8129, 2981, 1298, or the reverse
thereof. Each fourdigit number formed is evenly factored by 11.
Move template to any position on the Square, read in similar fashion.
B. In the top, lefthand corner of the REUT · REUT Square, highlight
the stepfigure 9317 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, lefthand corner of the REUT · REUT Square, highlight
the stepfigure 988122 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, lefthand corner of the REUT · REUT Square, highlight
the stepfigure 936174 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, lefthand 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, 293. On the
fourth ROW , immediately following the 3, highlight the three numbers
122, below the final 2, highlight the three numbers 817. The
stepfigure 988293122817 is a Transmagic Long number
_{9}

_{8}

_{8}


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B. Similarly, in the top row of the REUT · REUT Square, highlight the
three numbers 431, then highlight the top three numbers in the eighth
column , 293. Next highlight the 679, fourth row. Lastly, highlight
the numbers 817 in the eleventh column. The stepfigure 431293679817
is a Transmagic Long Number.
C. At the top of the first column highlight the three numbers 936.
Next, on the fourth row . highlight the numbers 643. On the fourth
column highlight the numbers 174. Lastly, on the seventh row,
highlight 467. The stepfigure 936643174467 is Transmagic.
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D.
Similarly, At the top of the sixth column highlight the three numbers
366. Next, on the third row, highlight the numbers 431. O n the
ninth column highlight the numbers 744. Lastly, on the sixth row,
highlight 679. The stepfigure 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 twelvedigit 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
79886431224
4. Pick
up the "6" 312246798864
679886431224
5.
Continue....
TEMPLATE
SUMS:
Elevens:
sum of the digits = 20 (onethird magic)
Thirty
sevens: sum of the digits = 30 (onehalf 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 Tableau1, the
Template value for one third Magic changes to become onesixth
Magic. The Template for onehalf magic changes to onefourth Magic.
MATHEMAGIC:
Transmagic
Squares contain layer upon layer of mathemagical
complexity. There are no limits to their scale, variety, or
digitlength, and they are infinite in number. Creating
Transmagic Squares that have the most possible common factors
in each column, row, diagonal, or brokendiagonal while maintaining the magic sum is the pursuit of
happiness.
Calculator Alligator!
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REUT · REUT 24
Transmagic Square
Magic sum 120  Magic Factors 7, 13
REUT · REUT TRANSMAGIC SQUARE,
ANALYSIS of the REUT · REUT
The REUT · REUT 24,
Transmagic Squares
by J. Frederic Teubner
© 20002023
J. Frederic Teubner
ALL RIGHTS RESERVED
