Expressions Using Letter-Numbers | FIO

Question 23

Observe the pattern below. How many squares will be there in Step 4, Step 10, Step 50? Write a general formula. How would the formula change if we want to count the number of vertices of all the squares?

Solution

Understand the Question

Each shape is formed by $1$ central square surrounded by $4$ identical arms.
In step $n$ , each arm consists of $n$ squares. Thus, the total number of squares is $S(n) = 4n + 1$ .
To find the number of distinct vertices, the central square contributes $4$ vertices, and each square added in an arm shares an edge and contributes $2$ new vertices, giving $V(n) = 4 + 8n = 8n + 4 = 2S(n) + 2$ .

Step 1 · Count Squares in the Initial Steps

Diagram 2

Counting the squares for each step:

Step 1: $1$ central square $+ 4$ arm squares: $1 + 4 = 5$
Step 2: $1$ central square $+ 4$ arms of $2$ squares each:

\begin{aligned} 1 + (4 \times 2) &= 1 + 8 \\[0.6em] &= 9 \end{aligned}

Step 3: $1$ central square $+ 4$ arms of $3$ squares each:

\begin{aligned} 1 + (4 \times 3) &= 1 + 12 \\[0.6em] &= 13 \end{aligned}

Step 2 · Find the General Formula for Number of Squares

Let $n$ be the step number and $S(n)$ be the total number of squares. Each step has $1$ central square and $4$ arms with $n$ squares each: $S(n) = 1 + (4 \times n)$ $S(n) = 4n + 1$

Step 3 · Calculate Number of Squares for Step 4, 10, and 50

Using the formula $S(n) = 4n + 1$ :

For Step 4 ( $n = 4$ ):

\begin{aligned} S(4) &= (4 \times 4) + 1 \\[0.6em] &= 16 + 1 \\[0.6em] &= 17 \end{aligned}

For Step 10 ( $n = 10$ ):

\begin{aligned} S(10) &= (4 \times 10) + 1 \\[0.6em] &= 40 + 1 \\[0.6em] &= 41 \end{aligned}

For Step 50 ( $n = 50$ ):

\begin{aligned} S(50) &= (4 \times 50) + 1 \\[0.6em] &= 200 + 1 \\[0.6em] &= 201 \end{aligned}

Step 4 · Find the General Formula for Number of Vertices

The central square provides $4$ vertices. Each attached square in the $4$ arms adds $2$ new vertices:

Step 1:

\begin{aligned} 4 + (4 \times 2) &= 4 + 8 \\[0.6em] &= 12 \end{aligned}

Step 2:

\begin{aligned} 4 + (4 \times 4) &= 4 + 16 \\[0.6em] &= 20 \end{aligned}

Step 3:

\begin{aligned} 4 + (4 \times 6) &= 4 + 24 \\[0.6em] &= 28 \end{aligned}

For Step $n$ , each of the $4$ arms has $n$ squares adding $2$ vertices each: $V(n) = 4 + (4 \times 2 \times n)$ $V(n) = 8n + 4$

Step 5 · Relate the Formula for Vertices to Squares

Expressing $V(n)$ in terms of $S(n)$ :

\begin{aligned} V(n) &= 8n + 4 \\[0.6em] &= 2 \times (4n + 2) \\[0.6em] &= 2 \times ((4n + 1) + 1) \\[0.6em] &= 2 \times (S(n) + 1) \\[0.6em] &= 2S(n) + 2 \end{aligned}

Therefore, the number of vertices is twice the number of squares plus $2$ .

Answer

• Step 4: $17$ squares, Step 10: $41$ squares, Step 50: $201$ squares • General formula for squares: $S(n) = 4n + 1$ • General formula for vertices: $V(n) = 8n + 4 = 2S(n) + 2$

Common Mistakes

Omitting the Central Square: Writing $S(n) = 4n$ instead of $4n + 1$ by forgetting the shared central unit square.
Overcounting Vertices: Multiplying the total squares by $4$ ( $4 \times S(n)$ ) without accounting for shared edges and vertices between adjacent squares.

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Q23