Square numbers and their patterns

If you can arrange a number of pebbles into a perfect square , same number of rows as columns , that number is called a square number or a perfect square. The list starts $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots$ and it has more surprises than you would expect.

Concept

A number $n$ is a perfect square if $n = k \times k$ for some whole number $k$. We write $k \times k$ as $k^2$ and read it as "$k$ squared". So $49 = 7 \times 7 = 7^2$ is a square; $50$ is not.

The geometric picture is the reason for the name. Lay out $k$ rows of $k$ pebbles each and you literally see a square. That is also why area is measured in square units: a $3 \text{ m} \times 3 \text{ m}$ room covers $9$ square metres because nine little $1 \text{ m} \times 1 \text{ m}$ tiles fit inside it.

Here is the first beautiful pattern. Write the odd numbers in a row and add them up step by step:

$1 = 1, \quad 1+3 = 4, \quad 1+3+5 = 9, \quad 1+3+5+7 = 16, \quad 1+3+5+7+9 = 25.$

Every running total is a perfect square. In general,

$1 + 3 + 5 + \dots + (2n-1) = n^2.$

The reason is geometric: each new odd number is a bent strip that wraps around the previous square to make the next bigger square.

The second useful pattern lets us jump from $n^2$ to $(n+1)^2$ without multiplying again. Since

$(n+1)^2 = n^2 + 2n + 1,$

we can find $11^2$ from $10^2$ as $100 + 2(10) + 1 = 121$. Or $21^2$ from $20^2$ as $400 + 41 = 441$. This is far faster than long multiplication.

A third pattern is the last-digit rule. Multiply any digit by itself and you get one of $0, 1, 4, 9, 6, 5, 6, 9, 4, 1$. So the last digit of a perfect square must be $0, 1, 4, 5, 6$, or $9$. Squares never end in $2, 3, 7, 8$. This little trick eliminates impossible answers in one glance , the number $1027$ ends in $7$, so it cannot be a square; you do not need to check.

Finally, perfect squares always have an odd number of factors. Every other number pairs its factors (like $12 = 1\times 12 = 2 \times 6 = 3 \times 4$, giving six factors). But for $36$, the pair $6 \times 6$ has the same number twice, so $36$ has the odd count $1, 2, 3, 4, 6, 9, 12, 18, 36$.

Worked examples

Example 1. Use the formula $(n+1)^2 = n^2 + 2n + 1$ to find $26^2$ from $25^2$.

• $25^2 = 625$.
• $26^2 = 625 + 2(25) + 1 = 625 + 51 = 676$.

Example 2. Without computing, can $2{,}358{,}752$ be a perfect square?

• The last digit is $2$.
• Perfect squares never end in $2$.
• So no, $2{,}358{,}752$ is not a perfect square.

Example 3. Add the first $50$ odd numbers.

• The sum of the first $n$ odd numbers is $n^2$.
• So the sum is $50^2 = 2500$.

Example 4. Find the missing number in the gnomon pattern: $1, 4, 9, ?, 25, 36$.

• These are $1^2, 2^2, 3^2, ?, 5^2, 6^2$.
• The missing square is $4^2 = 16$.

Try it yourself

1. Is $144$ a perfect square? If so, of what number?
2. Without multiplying, find $31^2$ from $30^2 = 900$.
3. Which of these cannot be a perfect square: $1089$, $4625$, $8908$, $76728$?
4. Add the first ten odd numbers in your head.
5. How many factors does $25$ have? List them.
6. The difference between two consecutive squares is $25$. What are the two numbers?
7. A square garden has area $169 \text{ m}^2$. What is its side length?
8. Why is the sum of two odd squares always even but never a square multiple of $4$? Try a few examples.

Activity

Square spiral. On graph paper start at a centre cell. Shade $1$ cell. Around it shade a bent strip of $3$ cells to form a $2\times 2$ square. Around that shade $5$ more cells to form a $3 \times 3$. Continue with strips of $7, 9, 11, \dots$ The strips you shade are exactly the odd numbers, and each completed square shows the identity $1 + 3 + 5 + \dots + (2n-1) = n^2$ in colour.