Magic
star...
Find a simple 'calculation-free' method to arrange
numbers from 1 to 16 at the nodes of the magic
octagram so that the constant sum of the numbers in every line is 34.
Variant
A
Fill in numbers 1 to 7 top down,
left to right...
Variant
B
Fill in numbers 16 to 10 down top,
right to left...
Shift
numbers 6-3 and 4-7 to the mirror
nodes in accordance with the symmetric
line.
Shift
numbers 10-13 and 14-11 to the
mirror nodes in accordance with
the symmetric line.
Place
number 16 next
to 1 as shown...
Place
number 1 next
to 16 as shown...
Consider
now the numbers in the blue discs.
The sum of two opposite numbers
should be equal to the sum of
the other pair of opposite numbers.
Hence:
3 + 16 = 7 + 12.
Place then the complement number
to 34 in the first column:
34 - (2 + 12 + 6) = 14.
Hence:
1 + 14 = 10 + 5.
Place then the complement number
to 34 in the last column:
34 - (11 + 5 + 15) = 3.
Finally,
filling the rest of numbers is
easy, since each line should add
to 34. Fill first the lines containing
3 numbers (blue numbers), then
progressively the other ones (green,
and orange numbers).
You've
done it! You
may find some interesting variants
of the magic octagrams above simply
by subtracting each number from
17; or by adding and subtracting
alternately the number 1 (or any
other integer) from the aligned
numbers. The rule is to keep the
basic numbers 1 to 16.
General
considerations
A
star polygon {p / q},
with p, q positive
integers, is a figure formed
by connecting with straight lines
every qth point
out of p regularly spaced
points lying on a circumference
(the number q is called
the density of the star polygon).
The octagram - also called '8-pointed
magic star' or 'star of Lakshmi'
(Ashtalakshmi) - is
a star polygon {8/2}.
A magic star is a star polygon
in which numbers placed at each
vertices and intersections, such
that the four numbers on each line,
sum to the same magic
constant. The magic constant
of a magic star contains the consecutive
integers 1 to 2n (n =
number of star points). The magic
constant of any n-pointed
magic star is M(n) = 4n +
2. Thus, the magic constant of
the octagram is: 4 x 8 + 2 = 34.
Any magic star can be made into
another magic star by complementing
each number of the original star
in turn. This is done by subtracting
each number from 2n +
1. In the case of the 8-pointed
magic star, which uses the numbers
1 to 16, you have to subtract each
number from 17 to obtain the new
number.
Interesting
properties of the octagram
a)
The sum of the numbers in the
blue discs is equal to the sum
of the numbers in the yellow
discs (see fig. 1 below,):
(B
+ H) + (E
+ K) = (C
+ I) + (F
+ L)
and also: B + H = E
+ K C +
I = F
+ L
If
A + D + G + J = M + N + O + P
= 34,
then, B + C =
G + J ; B + L =
N + O ; C + E =
O + P ; E + F =
A + J ; F + H =
M + P ; H + I =
A + D ; I + K =
M + N ; K + L =
D + G ;
b)
The sum of numbers placed in
any vertex and its adjacent nodes
(forming a small triangle) is
equal to the sum of numbers placed
in the opposite vertex and its
second subsequent nodes (forming
a larger triangle, see example
in fig. 2).
B
+ M + C = K
+ O + F
c)
Using method a) and b) you can
find any new magic octagram!
This
is not a blank page, just a page
about blank pages!
"The
blank page gives the right to dream"
- Gaston Bachelard
An intentionally
blank page is a page that
is devoid of content, and may be
unexpected. Usually these pages often
carry a notice such as "this
page intentionally left blank" (in
other languages: "questa pagina è stata
lasciata intenzionalmente vuota", "page
intentionnellement laissée
vierge", "esta página
ha sido expresamente dejada en blanco", "diese
Seite wurde absichtlich leer gelassen").
The phrase is a paradox: a self-refuting
meta-reference in that it falsifies
itself by its very existence on the
page in question. The reason for
these notices is to make it clear
to the reader that the blank page
is not the result of a printing error.
Obviously,
printing "this page intentionally
left blank" on a page is self-refuting,
since a blank page cannot have any
writing. Some organizations have realized
this and changed the blank page warning
into: "no test content on this
page".
Intentionally blank pages placed at the end of books are often used
to balance the folios which comprise the book. Often these pages are completely
blank with no such statement, or are used as "Notes" pages, serving
a practical purpose. In the case of telephone number directories, these pages
are often used to list important numbers and addresses.
Book publishers have also used stylized designs underneath the last
paragraph of a chapter to indicate that no other content is to be expected till
the next chapter, allowing for the possibility of blank pages without misunderstandings
by the readers...
An
intentional blank page on a PDF doc
(another
outstanding sample of
blank page)
Notable
variations
A
similar example to the intentionally
blank page is known to be used in electric
road warning signs, normally used to
indicate traffic problems down the
road. Rather than have the sign remain
blank when not in use, phrases such
as "this sign is currently out
of order" have been displayed.
This allows drivers to know that the
sign is still functional.
No
Warning Signs...
Another
example of self-referential
paradox... >>>
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