| This was not a challenge in the Naturally Mathematical Challenges, but
just came as an idea as a result of the hexagon-based challenge.
To find the solution, it really helps to look at simpler cases first.
If you do that, you will soon find that a remarkable pattern emerges.
Here, we'll summarise the number of hexagons of each side-length that
can be found in a larger figure. As we count the number of hexagons in a
hexagon of side n, we find that there is 1 of side n, 2 + 3 + 2 = 7 of
side n - 1, and so on.
| Largest |
|
________ |
|
Smallest |
|
|
|
... |
n |
|
|
3 |
... |
n+1 |
|
2 |
4 |
... |
... |
| 1 |
3 |
5 |
... |
2n-1 |
|
2 |
4 |
... |
... |
|
|
3 |
... |
n+1 |
|
|
|
... |
n |
After examining the cases for n = 1, 2 and 3, it seems reasonable to
make the hypothesis that Hn= n3, i.e. that
the number of regular hexagons in a hexagon of side n is n3.
H1=1 is true. If Hn= n3 is assumed
true, then
This
reduces to Hn+1=(n+1)3.
Hence
by mathematical induction, Hn= n3
is true for all n.
|
