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corner top left Previous Puzzles of the Month + Solutions  
September-October 2004  

thinking man
logo puzzle of the month 1 Puzzle #99
Quiz/test #9 logo pzm 2
logo pzm 3 W-kammer #9
   Enjoy solving Archimedes' Lab™ Puzzles!

triangle-square-circle 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!
5 to 1 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).
triangular pieces solution
Finally, you can recompose a larger equilateral triangle by adjusting all the pieces around the 7th triangle...

Previous puzzles of the month...

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circle-triangle Quiz #9 TOP
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 =

a) b) c) result

Wunderkammer #9 TOP

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.

4x4 Latin square

  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.

4x4 Graeco-Latin 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.

graeco-latin square

  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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Math Gems

1/4 + (1/4)2 + (1/4)3 + (1/4)4 + ... = 1/3
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