How many regular hexagons can you find in this figure?

What if there were 7 triangles on each side of the figure, instead of 3?

We can work this out by looking at what happens when we increase the size of the figure by one triangle, and count the numbers of new heaxagons we can find.

Let's increase the figure to 4 triangles on each side (new triangles shown darker):

There are 3 new hexagons with a side length of 1, for each of the six sides of the figure:

There are 2 new hexagons with a side length of 2, for each of the six sides of the figure:

There is 1 new hexagon with a side length of 3, for each of the six sides of the figure:

Finally, we count the outer hexagon (side length 4):

The centre of each hexagon we found was marked with a dot. Plotting all the dots we get:

We've found that the number of new hexagons we get by increasing the figure from n to n+1 triangles per side is given by the number of spots in the hexagon figure which has n+1 spots on each side

This number is called the "hexagon centred number". The series goes 1, 7, 19, 37, 61...

We now just need to check that the figure with 1 triangle per side contains a single hexagon.. ah yes, that's true. To add a second triangle to the figure we add the second hexagon centred number (7) to find the number of hexagons possible. Adding each triangle to the side of the figure adds the next hexagon centred number - we can arrange these centred hexagons into a pyramid. We have a sequence called "pyramidal hexagon centred numbers" whose nth term is simply the cube of n.

So, for figures with 1, 2, 3, 4, 5, 6, 7.. triangles per side there are 1, 8, 27, 64, 125, 216, 343.. hexagons.

Very pleasing! I thank Hexagonia for drawing the problem to my attention. The solution method is my own. Their solution is quite nice too.

page date: 22Mar04.      I enjoy correspondence stimulated by this site. You can contact me here.