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The Radcliffe Shape

To see why there is only one Radcliffe shape or H-hexagon that can be magic, we need to find formulae for two features of the figure:
  1. How many rows are there in a shape of side 1, 2, 3, ..., n
  2. How many hexagons are there in a shape of side 1, 2, 3, ..., n

The first question can be answered by making a table of values and spotting what the underlying pattern is:

Side Rows
1 1
2 3
3 5
n 2n - 1
The following diagram shows why this formula works

Moving from a 2-side to a 3-side we have to add 2 more rows.

To find a formula for the number of small hexagons in an n-sided hexagon, look at the following pattern

Side # of hexagons
1 1
2 2 + 3  + 2 = 7
3 3 + 4 + 5 + 4 + 3 = 19
4 4 + 5 + 6 + 7 + 6 + 5 + 4 = 37

The pattern is clear, and there's a nice way to see how to turn it into a formula. For example, taking the expression for the 4-side hexagon, we can rewrite this as

0 + 1 + 2 + 3 + 2 + 1 + 0

4 + 4 + 4 + 4 + 4 + 4 + 4

That is, we add up the numbers 1 to  n - 1 twice and subtract n - 1, then we add n to itself n - 1 times. After a bit of algebra, this give the formula below:

Side # of hexagons
n 3n2 - 3n + 1

This means that the numbers 1 up to 3n2 - 3n + 1 have to be arranged into 2n - 1 rows, each having the same total. This can only be done if the total of all these numbers is itself divisible by 2n - 1. Trigg showed that this can only happen when 5/(2n-1) is an integer, which means that n = 1 (the trivial case of one hexagon) and n = 3 (the Radcliffe hexagon) are the only values of n for which a magic hexagon can be found. And the only possibility turns out to be the one shown on the previous page.

End of story? We don't think so!