Ultra Magic squares – Walter Trump
In 2002 Walter Trump investigated unusual
magic squares which he decided to call ultra-magic. Most of the wording, and
all of the examples shown here is from material condensed from several emails
and attached spreadsheets sent to me in February 2002.
Some of this work was in collaboration with George Chen.
Square a.
is one of 17 order-7 Walter found that are all pandiagonal, center
symmetrical (=associative), and contain two magic 3x3 arrays
These are squares 04347674 and 06969925 from Trump's database of
20,190,684 squares.
Squares
b., c.,
and d. are 1 of 11 magic squares of
order 7 with following features
- pandiagonal
- center symmetric
(associative)
- bent main-diagonals sum
up to 7 x 25 (= magic sum)
- in each 4x4 quadrant
(overlapping) the diagonals sum up to 4x25 (square b.)
-
Several sets of three cells sum up to 3 x 25 (squares c. and d.)
There are 96 magic
squares of order 7 with the following features
- center symmetric
(associative)
- pandiagonal
- the sum of the squares of
all numbers in the middle row is the same as in the middle column
- the sum of the squares of
all numbers in the left diagonal is the same as in the right diagonal
- the sum of the third
powers of all numbers in the middle row is the same as in the middle column
- the sum of the third
powers of all numbers in the left diagonal is the same as in the right
diagonal
Mathematical statement
(easy to proof):
-
If a magic square
fulfils the conditions 1 and 3, then it also owns the features 5
-
If a magic square
fulfils the conditions 1 and 4, then it also owns the features 6
-
If a square is center
symmetric, then two rows or two columns symmetrically located to the center
have equal squares-sums and equal third-powers-sums
The complete set (96
squares) consist of 16 sets, each containing 6 squares that belong to the same
transformation-group. From each of the 16 groups I show only one square.
|
There are 11
magic squares of order 7 with the following features
-
pandiagonal
-
center
symmetric (associative)
-
letter H as
quadrant magic pattern
-
the squares
of the numbers in the H-patterns have equal sums
|
 |
There are 20 magic
squares of order 7 with the following features
-
pandiagonal
-
center symmetric (associative)
-
letter H as quadrant magic pattern
-
a magic 3 by 3 array is inlaid (red numbers)
-
Additional features of square 14
-
the sum of the squares of all numbers in the
middle row is the same as in the middle column
-
the sum of the cubes of all numbers in the
middle row is the same as in the middle column
|
There are 30
magic squares of order 7 with following the features
The numbers
in each tile sum up to 3x25 = 75
Square 00384300
|
 |
These 4 types of ultra letter squares
are all pandiagonal center symmetric (associated) magic
|
a. the sum of the 10 numbers in
each letter A sum to 250. The squares of these numbers sum to 8362.
There are 11 order-7 magic squares with these features.
b. the sum of the 7 numbers in each
letter C and H sum to 175. The squares of these numbers sum to 5721.
There is only 1 magic square with these features?
Squares c. and
d. are shown twice because the patterns overlap.
c. the sum of the 10 numbers in
each letter A sum to 250. The squares of these numbers sum to 7622.
There are 5 magic squares with these features.
d. the sum of the 10 numbers in each
letter A and W sum to 250. There are 60 magic squares with these
features. |
|
Walter Trump has a web page on
this subject with much more detailed information at
http://www.nefkom.net/trump/magic-squares
| 
