Partitioning A Hexagon

Every picture can be subdivided into six equal regions which contain exactly $n^{2}$ triangles each; thus, the ninth picture will have $6*9^{2} = 486$ triangles in total.

Pattern #1 includes 6 triangles.

Pattern #2 includes 24 triangles.

Pattern #3 includes 54 triangles.

We can notice right off the bat that all three of these numbers are divisible by 6.

When we divide each number by 6, we get the following sequence: 1, 4, 9

1, 4, and 9 are $1^2, 2^2$ , and $3^2$ , respectively.

Therefore, we can conclude that the pattern can be described by the following equation: $x=6n^2$

Since we're looking at the ninth item in the sequence: $(6)9^2=486$

Another solution is to find the series of how many tiles are added to the outside as each layer is added.

By observation the first three layers have $6\cdot 1$ , $6 \cdot 3$ , and $6 \cdot 5$ added to the outside of the previous level (6 sides to the shape times $2i-1$ triangles added per side on the $i$ th later), leading to to the total tiles at each level being the sum of tiles added up to that point:

$\sum_{i=1}^n 6 \cdot (2i - 1)$ which is equivalent to $6 \sum_{i=1}^n (2i - 1)$ where $n$ is the number of layers.

Using the formula for the sum of a series, $\sum_{i=1}^n a_i = \frac{n}{2}(a_1 + a_n)$ , this gives:

$6 \sum_{i=1}^9 (2i - 1)$ = $6 \cdot (\frac{9}{2}(1+17)) = 6 \cdot 9 \cdot 9 = 6 \cdot 81 = \boxed{486}$

Each picture is made of 6 larger triangles, which each equal to $n$ squared. Or, $n$ squared x 6. 9 squared x 6 is 486 triangles total.

A hexagon can be cut into 6 triangles. I will refer to these as sectors to minimize confusion. The pattern goes that you square the placement of the hexagon (first hexagon = 1, second hexagon =2, etc) to find the number of triangles within a sector of the hexagon. So, for example, the first hexagon can be broken up into 6 triangles. It's placement is first, so you square 1. 6*(1^2) = 6 triangles (first shape)

The pattern continues:

6 (2^2)=24 triangles (second shape).... 6 (3^2) triangles = 54 triangles (third shape)...

To answer the question:

6*(9^2) triangles = 486 triangles

486 is the correct answer

I observed the first 3 iterations:

6,24,54 and came up with an easy solution:

suppose iteration = n, so we can make a function:

P(n)=6*n^2

6*9^2=486

This question is easy, how come only 46% got it right?

the hexagon can be dissected into 6 big triangles, the $n^{th}$ triangle have 2 sequence of triangles, the upside one $(1+2+...+n)$ and the downside one $1+2+...+(n-1)$ .

so, total small triangles will be: $6*(\frac{n * (n+1)}{2} + \frac{n * (n-1)} {2})$ = $6*\frac{n*(n^2-1)}{2}$

by substituting $n = 9$ , you will get $\boxed{486}$

I got the solution to write yay, let me explain how, you can notice that the hexagon contains 6 different (big)triangles for each triangle you can calculate the (little)triangles by noticing that it's the sum of the first n odd numbers $(n =1 -> S=1, n = 2 ->S = 1+3=4, n=3 ->S=1+3+5=9 ..etc)$ if you would notice the sum(S) always equals $n^2$ there is a simple proof for this find it any were online you might also know this if you study number theory but if you couldn't notice the pattern you could have easily calculated the sum of the first n odd numbers.

• at $n = 9$ (ie the 9th pattern)
• no. small triangles in a big triangle $(S) = n^2 = 9^2 = 81$
• no. small triangles in the hexagon = no. big triangles in a hexagon $* S = 6*81 = 486$

a challenge that would be interesting: what is the total number of triangles (any size) in the Nth figure

hint: start by solving it for the 9th figure

The formula to get the number of little triangles on the $n^{th}$ figure is $6 * n^2$ .

Plugging $9$ in place of $n$ gives $6 * 9^2 = 486$ , which is our answer.