#
Problem
of the Week

Problem
A and Solution

Messy
Mosaic

## Problem

A pattern of figures is made from square blocks. Here are the first
four figures of the pattern:

Describe a *pattern rule* for the pattern.

Using your pattern rule from part (a), what would Figure \(10\) be in the pattern?

Using your pattern rule from part (a), how many blocks are in
Figure \(100\)?

## Solution

Here is one possible pattern rule. Each figure has a single
column that is one block high, followed by columns that are three blocks
high. Each figure of the pattern has one more three-block column than
the previous figure.

Using the pattern we described in part (a), we could draw
pictures of the fifth through ninth figures before drawing the tenth
figure. However, we notice that Figure \(1\) has \(1\) three-block column, Figure \(2\) has 2 three-block columns, Figure \(3\) has \(3\) three-block columns, and Figure \(4\) has \(4\) three-block columns. From this, we can
extrapolate that Figure \(10\) has
\(10\) three-block columns.

Using the same logic as in part (b), there will be \(100\) three-block columns in Figure \(100\), giving a total of \(3 \times 100 = 300\) blocks. When we
include the first column in our count, this means there are \(301\) blocks in Figure \(100\).

**Teacher’s Notes**

This problem provides a visual example of an *arithmetic
sequence*. An arithmetic sequence is a sequence of numbers in which
each number after the first is obtained from the previous number by
adding a constant, called the common difference. In this problem, the
common difference is \(3\).

The mathematical expression we can use to describe the general form
of the \(n^{\text{th}}\) term in an
arithmetic sequence is \[a_n = a_1 + (n -
1)d\]

where \(a_1\) is the first term of
the sequence, \(d\) is the common
difference between pairs of numbers in the sequence, and \(a_n\) is the \(n^{\text{th}}\) term of the sequence.

In this problem, the first term of the sequence is \(4\) and the common difference is \(3\), so the mathematical expression for the
\(n^{\text{th}}\) term of the sequence
is \[a_n = 4 + (n - 1)(3)\]

We can use this formula to calculate the number of blocks in Figure
\(10\): \[a_{10} = 4 + (10 - 1)(3) = 4 + (9)(3) = 4 + 27 =
31\]

We can also use this formula to calculate the number of blocks in
Figure \(100\): \[a_{100} = 4 + (100 - 1)(3) = 4 + (99)(3) = 4 +
297 = 301\]