| In a book published in 1921, New Mathematical Pastimes, P. A. MacMahon
suggested a novel form of tessellation problem, based on what have come
to be known as MacMahon's Triangles:
An equilateral triangle is divided into three parts, which can be
numbered or coloured, as the fancy takes you.
Problem 1
Find a formula to express the number of distinct (i.e. non-congruent)
MacMahon triangles that can be formed with 1, 2, 3, ..., n numbers or
colours.
Suggestion: Start with n = 1, 2 and even 3 and see if a
pattern starts to emerge for you.
Problem 2
When four numbers or colours are used, find a way in which to arrange
the MacMahon triangles into a T-hexagon such that:
- The numbers/colours round the edge of the T-hexagon are all the
same.
- The numbers/colours along an edge that is common to 2 triangles
are the same.
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allowed |
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not allowed |
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