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eureka!!!
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corner top left Previous Puzzles of the Month + Solutions  
September-October 2005  

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logo puzzle of the month 1 Puzzle #103
Quiz/test #13 logo pzm 2
logo pzm 3 W-kammer #13
   Enjoy solving Archimedes' Lab™ Puzzles!

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triangle-square Puzzle #103
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.
italiano/francais
magic octagon
solution
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Variant A
Fill in numbers 1 to 7 top down, left to right
...
octagram 1
Variant B
Fill in numbers 16 to 10 down top, right to left
...
octagram 2
Shift numbers 6-3 and 4-7 to the mirror nodes in accordance with the symmetric line.octagram 3 Shift numbers 10-13 and 14-11 to the mirror nodes in accordance with the symmetric line.octagram 4
Place number 16 next to 1 as shown...
octagram 5
Place number 1 next to 16 as shown...
octagram 6

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.

octagram 7
Hence: 1 + 14 = 10 + 5.
Place then the complement number to 34 in the last column:
34 - (11 + 5 + 15) = 3.

octagram 8
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).
octagram 9 octagram 10

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.

 

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General considerations TOP

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!

fig. 1octagram 11 fig. 2octagram 12

See also:
small square Latin and Graeco-Latin Squares
small square Latin Magic Square

 

Please report any error, or misspelling. Thanks!
© Sarcone & Waeber, Archimedes' Lab


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circle-triangle Quiz/Test #13 TOP
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Quick Math Questions
Find the difference between 0.10 and 0.7... If C is larger than A and B is cheaper than A, may A be smaller than B, and more expensive than C? If C is larger than A and B is cheaper than A, may A be larger than B, and cheaper than C?
complete
a) Sure
b) Nope
a) Sure
b) Nope


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Wunderkammer #13 TOP
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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
blank page
(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... >>>

no warning signs

External links
small square Intentionally-left-blank Project
small square Please-wait Project

Related Book
book title

Everything Men Know About Women
by Knott Mutch
A hilarious, ice-breaker of a blank book that just keeps on pleasing, and selling, year after yeary.

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