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Previous Puzzles of the Month + Solutions
September-October 2005

 Puzzle #103 Quiz/test #13 W-kammer #13
Enjoy solving Archimedes' Lab™ Puzzles!

 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.
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 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! fig. 1 fig. 2 See also: Latin and Graeco-Latin Squares Latin Magic Square Please report any error, or misspelling. Thanks! © Sarcone & Waeber, Archimedes' Lab

Previous puzzles of the month...
 August 98: the irritating 9-piece puzzle September 98: the impossible squarings October 98: the multi-purpose hexagon November 98: Pythagora's theorem December 98: the cunning areas January 99: less is more (square roots) February 99: another square root problem... March 99: permutation problem... April 99: minimal dissections July 99: jigsaw puzzle August 99: logic? Schmlogic... September 99: hexagon to disc... Oct-Nov 99: curved shapes to square... Dec-Jan 00: rhombus puzzle... February 00: Cheeta tessellating puzzle... March 00: triangular differences... Apr-May 00: 3 smart discs in 1... July 00: Funny tetrahedrons... August 00: Drawned by numbers... September 00: Leonardo's puzzle... Oct-Nov 00: Syntemachion puzzle... Dec-Jan 01: how many squares... February 01: some path problems... March 01: 4D diagonal... April 01: visual proof... May 01: question of reflection... June 01: slice the square cake... July 01: every dog has 3 tails... Aug 01: closed or open... Sept 01: a cup of T... Oct 01: crank calculator... Nov 01: binary art... Dec 01-Jan 02: egyptian architecture... Feb 02: true or false... March 02: enigmatic solids... Apr 02: just numbers... May 02: labyrinthine ways... June 02: rectangle to cross... July-Aug 02: shaved or not... Sept 02: Kangaroo cutting... Oct 02: Improbable solid... Dec-Jan 03: Hands-on geometry Feb-Mar 03: Elementary my dear... Apr-May 03: Granitic thoughts June-July 03: Bagels... September 03: Larger perimeter... Oct-Nov 2003: square vs rectangle Dec-Jan 04: curvilinear shape... February 04: a special box March 04: magic 4 T's... April O4: inscribed rectangle May 04: Pacioli puzzle... June 04: pizza's pitfalls October 04: Odd triangles February 05: Same pieces May-June 05: stairs to square July-August 05: cheese!
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 Quiz/Test #13
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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

 Wunderkammer #13
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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

(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... >>>

 External links Intentionally-left-blank Project Please-wait Project Related Book 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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