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Previous Puzzles of the Month + Solutions
September-October 2004

 Puzzle #99 Quiz/test #9 W-kammer #9
Enjoy solving Archimedes' Lab™ Puzzles!

 Puzzle #99
Odd triangles
It is more difficult to cut and rearrange 5 small triangles to form a larger one than 7 small triangles...
According to the example below, cut 7 equilateral triangles with just one straight cut and then rearrange all the pieces (without overlappings) to make another equilateral triangle!

(click the image to enlarge it)
Arrange 6 of the 7 triangles as shown below, then cut them with just one straight cut (dotted line).

Finally, you can recompose a larger equilateral triangle by adjusting all the pieces around the 7th triangle...

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
Back to puzzle-of-the-month page

 Quiz #9
 Test your visual attention online 1. Mary's Father has 5 daughters: • Chacha, • Cheche, • Chichi, • Chocho... Find the name of the fifth one! 2. Which one of these structures is impossible? 3. Add up quickly: 1000 + 40 + 1000 + 30 + 1010 + 20 = complete a) b) c) result

 Wunderkammer #9

The Latin and Graeco-Latin Squares

Latin squares and magic squares are the first matrices studied. The Swiss mathematician Leonhard Euler first investigated square arrays in which symbols appeared once in each row and column, and named them 'Latin squares' since he used letters of the Latin alphabet. Actually, a Latin square is an n x n table which can be filled with n different 'symbols' (letters, colors, shapes, objects, etc.) in such a way that each symbol occurs exactly once in each row and exactly once in each column.

 A B C D B A D C C D A B D C B A

Two Latin squares of order n are said to be orthogonal if one can be superimposed on the other, and each of the n2 combinations of the symbols (taking the order of the superimposition into account) occurs exactly once in the n2 cells of the array. Such pairs of orthogonal squares are often called Graeco-Latin squares since it is customary to use Latin letters for the symbols of one square and Greek letters for the symbols of the second square.

 A α B γ C δ D β B β A δ D γ C α C γ D α A β B δ D δ C β B α A γ

Here below is an other example of a color Graeco-Latin square of order 10.

In the diagram above, the two sets of 'colors/symbols' are identical (there are 10 different colors in all). The larger squares constitute the Latin square, while the inner squares constitute the Greek square. Every one of the 100 combinations of colors (taking into account the distinction between the inner and outer squares) occurs exactly once. Note that for some elements of the array the inner and outer squares have the same color, rendering the distinction between them invisible.
Graeco-Latin squares have applications in the design of scientific and pharmacological experiments, and they are interesting as mathematical objects.

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