Magic Hexagon of Triangles

Can we arrange the numbers 1 to 24 in 24 triangular cells forming a hexagon, so that all rows (horizontal and diagonal, 12 in total) add up to the same number?

We certainly can... There are slightly short of 60 million ways to do this.

The exact number is 59,674,527, which was found by computer program.

You'll forgive me if I don't list them all. This is an example, picked at random:-

Here are a number of results which hold for all these magic hexagons:-

Blue region adds up to 75:-

Blue region adds up to 150:-

Blue and green regions add up to the same:-

Blue region adds up to (75 + green region):-

Here is a page on which these results are derived.

These results can also be derived visually, and extended to larger hexagons. See this article by John Baker and myself for more details. John and I have done some very enjoyable work together on these magic hexagons.

One of our nice discoveries is what we call the Odd-Even property. For any of these magic hexagons, and indeed for any hexagon with constant row sums, the sum of the numbers in the triangles with apex pointing up is the sum of the numbers in the triangles with apex pointing down. Visually, the blue triangles sum to the green triangles:

Using this result we have shown that magic hexagons are physically balanced about their diagonals - a property we call magic moments!

In other words, if each triangle weighs the same as the number it contains, the hexagon will balance on a wire across any diagonal. In addition it will balance on a wire across mid-points of opposite sides

This even works for the magic hexagon of hexagons, and for magic squares.

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Furthermore it works as long as the row sums are constant, even if the numbers used are not sequential. For instance it works for this hexagon, which has several 3s, zeroes and a negative number.

John Baker (of Hexagonia and Natural Maths) and I have published an article on this in the July 2006 Mathematical Gazette. Our collaboration and discoveries together have been a total joy.

page date: 22Aug04.      I enjoy correspondence stimulated by this site. You can contact me here.