Question 2

The Sierpinski Triangle is a special shape called a fractal. We make it by starting with a triangle and repeatedly removing its middle part.

Step 1 — Counting the holes

Let us count the number of empty, uncolored triangular spaces, which we call holes, at each step.

At Step 0, we start with one solid triangle. There are no empty spaces inside it. $\text{Number of holes at Step 0} = \mathbf{0}$

At Step 1, we remove the middle inverted triangle from the original triangle. This creates one hole. $\text{Number of holes at Step 1} = \mathbf{1}$

At Step 2, we take each of the three solid triangles from Step 1. From each of these three triangles, its own middle inverted triangle is removed. This adds 3 new holes. We already had 1 hole from Step 1. $\text{Number of holes at Step 2} = (\text{holes from Step 1}) + (\text{new holes added})$ $= 1 + 3$

$\boxed{\mathbf{4} \text{ holes}}$

At Step 3, we continue the process. There are 9 solid triangles remaining from Step 2. Each of these 9 triangles will have its middle part removed. This adds 9 new holes. We already had 4 holes from Step 2. $\text{Number of holes at Step 3} = (\text{holes from Step 2}) + (\text{new holes added})$ $= 4 + 9$

$\boxed{\mathbf{13} \text{ holes}}$

At Step 4, we repeat the process again. There are 27 solid triangles remaining from Step 3. Each of these 27 triangles will have its middle part removed. This adds 27 new holes. We already had 13 holes from Step 3. $\text{Number of holes at Step 4} = (\text{holes from Step 3}) + (\text{new holes added})$ $= 13 + 27$

$\boxed{\mathbf{40} \text{ holes}}$

We can see a pattern for the number of new holes added at each step. At Step 1, we added $3^0 = 1$ hole. At Step 2, we added $3^1 = 3$ holes. At Step 3, we added $3^2 = 9$ holes. At Step 4, we added $3^3 = 27$ holes. So, at Step $n$ (for $n \ge 1$), we add $3^{n-1}$ new holes. The total number of holes at Step $n$ is the sum of all new holes added up to that step. $\text{Number of holes at Step n} = 3^0 + 3^1 + 3^2 + \dots + 3^{n-1}$

$\boxed{\text{Number of holes at Step n} = 3^0 + 3^1 + 3^2 + \dots + 3^{n-1}}$

Step 2 — Counting the remaining triangles

Let us count the number of solid, colored triangles that remain at each step.

At Step 0, we start with one large solid triangle. $\text{Number of remaining triangles at Step 0} = \mathbf{1}$

<DIAGRAM: A single solid purple triangle labeled "Step 0".>

At Step 1, the middle part is removed. This leaves three smaller solid triangles at the corners. $\text{Number of remaining triangles at Step 1} = \mathbf{3}$

At Step 2, each of the three solid triangles from Step 1 is further divided. Each of them becomes three smaller solid triangles. So, we multiply the previous number of triangles by 3. $\text{Number of remaining triangles at Step 2} = 3 \times (\text{triangles from Step 1})$ $= 3 \times 3$

$\boxed{\mathbf{9} \text{ triangles}}$

At Step 3, each of the nine solid triangles from Step 2 is further divided into three smaller solid triangles. $\text{Number of remaining triangles at Step 3} = 3 \times (\text{triangles from Step 2})$ $= 3 \times 9$

$\boxed{\mathbf{27} \text{ triangles}}$

At Step 4, each of the twenty-seven solid triangles from Step 3 is further divided into three smaller solid triangles. $\text{Number of remaining triangles at Step 4} = 3 \times (\text{triangles from Step 3})$ $= 3 \times 27$

$\boxed{\mathbf{81} \text{ triangles}}$

We can see a pattern for the number of remaining triangles. At Step 0, we have $3^0 = 1$ triangle. At Step 1, we have $3^1 = 3$ triangles. At Step 2, we have $3^2 = 9$ triangles. At Step 3, we have $3^3 = 27$ triangles. At Step 4, we have $3^4 = 81$ triangles. So, at Step $n$, the number of remaining triangles is $3^n$.

$\boxed{\text{Number of remaining triangles at Step n} = 3^n}$

Answer

(i) Number of holes: Step 0: 0 hole Step 1: 1 hole Step 2: 4 holes Step 3: 13 holes Step 4: 40 holes Step n: $3^0 + 3^1 + 3^2 + \dots + 3^{n-1}$ holes

(ii) Number of triangles that remain: Step 0: 1 triangle Step 1: 3 triangles Step 2: 9 triangles Step 3: 27 triangles Step 4: 81 triangles Step n: $3^n$ triangles