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...
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
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.
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.
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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