Odd Numbers Build Squares, math that surprises you lesson

Add the odd numbers in order. The totals are $1$ , then $1 + 3 = 4$ , then $1 + 3 + 5 = 9$ , then $1 + 3 + 5 + 7 = 16$ , and every one of them is a square number. Does this go on forever, and why? This page answers with a picture of dots in which each odd number is exactly the L-shaped border that one square needs to become the next, explains why that picture proves the pattern for every $n$ , and then reads it backward to reach the difference of squares, with the conditions under which each picture applies.

Key terms

Sum of odd numbers

Definition For every whole number $n \ge 1$ , the sum of the first $n$ odd numbers is $n^2$ ,

1 + 3 + \cdots + (2n - 1) = n^2.

The $k$ th odd number is $2k - 1$ , so the last term of the sum is $2n - 1$ . To see why the sum is $n^2$ , arrange $n^2$ dots in a square of $n$ rows and $n$ columns and split it into $n$ layers that grow out of the bottom left corner. For each $k$ from $1$ to $n$ , layer $k$ is the top row together with the right column of the $k$ by $k$ square of dots in the bottom left corner. That row and that column each have $k$ dots and share their corner dot, so layer $k$ has $k + k - 1 = 2k - 1$ dots. Each dot of the big square lies in exactly one layer, so adding the layers counts every dot once, and the total is $n^2$ . An L-shaped border of this kind is called a gnomon.

In plain words Each odd number is exactly the L-shaped border that turns one square of dots into the next larger square.

Example In the $4$ by $4$ square below, each dot is replaced by the number of its layer.

\begin{array}{cccc} 4 & 4 & 4 & 4 \\ 3 & 3 & 3 & 4 \\ 2 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}

The digit $1$ appears once, $2$ three times, $3$ five times and $4$ seven times, and together they fill all $16$ places, so $1 + 3 + 5 + 7 = 16 = 4^2$ . The fourth layer is a row of four across the top and a column of four down the right side, sharing the top right place, so it has $4 + 4 - 1 = 7$ dots. A fifth layer would have $5 + 5 - 1 = 9$ dots and give $16 + 9 = 25 = 5^2$ .

Square numbers

Definition A square number, or perfect square, is a number $n^2 = n \times n$ for a whole number $n \ge 0$ . The square numbers are $0, 1, 4, 9, 16, 25, 36, \ldots$ , and $n^2$ counts the dots in a square array of $n$ rows and $n$ columns. A positive whole number is a square number exactly when that many dots can be arranged in a full square with none left over. For a length $x \ge 0$ that need not be a whole number, $x^2$ is the area of a square with side $x$ . The gaps between consecutive square numbers are $1, 3, 5, 7, \ldots$ , the odd numbers in order.

In plain words A square number is a whole number multiplied by itself, the number of dots in a complete square arrangement.

Example The number $49 = 7 \times 7$ is a square number. The number $50$ is not, because $7^2 = 49 < 50 < 64 = 8^2$ , so $50$ dots make a $7$ by $7$ square with one dot left over and are too few for an $8$ by $8$ square. Between the square numbers $0, 1, 4, 9, 16$ the gaps are $1, 3, 5, 7$ , which are the four layers of the $4$ by $4$ square above.

Proof without words

Definition A proof without words is a picture arranged so that a reader can see why a statement is true and could write out the complete argument from it. A picture counts as a proof only when its construction works in every case the statement covers, not just in the cases drawn, and when every piece has a precise job, with each dot counted exactly once, or with areas cut and moved without gaps, overlaps or stretching. A dot picture proves a statement about whole numbers. An area picture proves a statement about lengths and areas, and only for the lengths it can draw, which are never negative.

In plain words The picture is the argument, as long as each part is accounted for exactly and the same construction works at every size.

Example The layered square is a proof without words of the sum of odd numbers, because the layer that turns a $k - 1$ by $k - 1$ square into a $k$ by $k$ square always has $k + k - 1$ dots, for every $k \ge 2$ , not only for the layers drawn. A picture that only looks right is not enough. In the missing square puzzle the same four pieces seem to fill the same triangle twice, once with an empty square. The long edge of the figure is not straight. It is made of two slopes, $3/8$ and $2/5$ , and the two figures differ in area by exactly one square unit.

Difference of squares

Definition For all real numbers $x$ and $y$ ,

x^2 - y^2 = (x - y)(x + y).

Algebra proves it in every case, since

\begin{aligned} &(x - y)(x + y) \\[0.4em] &\quad = x^2 + xy - yx - y^2 \\[0.4em] &\quad = x^2 - y^2. \end{aligned}

When $x \ge y \ge 0$ there is also an area proof. From a square of side $x$ remove a square of side $y$ at one corner, which leaves an L-shaped region of area $x^2 - y^2$ . Cut that region into a rectangle $x - y$ wide and $x$ tall and a rectangle $y$ wide and $x - y$ tall. Turn the second piece a quarter turn, so that it is $x - y$ wide and $y$ tall, and set it on top of the first. The two pieces now form one rectangle $x - y$ wide and $x + y$ tall, and cutting and moving did not change the area. With $x = n$ and $y = n - 1$ the identity gives $n^2 - (n - 1)^2 = 2n - 1$ , which is the newest layer of the dot picture.

In plain words Taking a small square out of the corner of a big square leaves an L-shape, and the L-shape can be cut and rearranged into a rectangle whose sides are the difference and the sum of the two side lengths.

Example Take $x = 7$ and $y = 3$ . Removing the $3$ by $3$ square from the $7$ by $7$ square leaves $49 - 9 = 40$ square units, in a $4$ by $7$ piece of area $28$ and a $3$ by $4$ piece of area $12$ . Turned and stacked, they form a $4$ by $10$ rectangle, and $4 \times 10 = 40$ . With $x = 10$ and $y = 9$ the identity gives $10^2 - 9^2 = 1 \times 19 = 19$ , the newest layer of a $10$ by $10$ square of dots. With $x = 50$ and $y = 1$ it helps with mental arithmetic,

\begin{aligned} 51 \times 49 &= 50^2 - 1^2 \\[0.4em] &= 2499. \end{aligned}

With $x = 3$ and $y = 7$ the area picture cannot be drawn, because the square of side $7$ does not fit inside the square of side $3$ , but the algebra still holds, since $9 - 49 = -40$ and $(3 - 7)(3 + 7) = -40$ .

Check yourself

Find $1 + 3 + 5 + \cdots + 99$ . How many terms does the sum have?

Show the answer

The $n$ th odd number is $2n - 1$ , and $2n - 1 = 99$ gives $n = 50$ . So the sum has $50$ terms and equals $50^2 = 2500$ . A common slip is to use the last term, $99$ , as $n$ , which would give $99^2 = 9801$ . A quick check pairs the first and last terms, $1 + 99 = 3 + 97 = \cdots = 100$ , and the $50$ terms make $25$ such pairs, so the sum is $25 \times 100 = 2500$ .

Change the rule and add the even numbers instead, $2 + 4 + \cdots + 2n$ . Are the totals still square numbers? What shape do the dots make?

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The totals are $2, 6, 12, 20, 30, \ldots$ , and none of them is a square number. They equal $n(n + 1)$ , the number of dots in a rectangle of $n$ rows and $n + 1$ columns. To grow a rectangle of $n - 1$ rows and $n$ columns into one of $n$ rows and $n + 1$ columns, add a row of $n + 1$ dots across the top and a column of $n$ dots down the right side. They share the top right dot, so the new border has $(n + 1) + n - 1 = 2n$ dots, which is the $n$ th even number. The rectangle is never a square, because

n^2 < n(n + 1) < (n + 1)^2

for every $n \ge 1$ , so $n(n + 1)$ lies strictly between two neighboring square numbers. Halving the totals also gives $1 + 2 + \cdots + n = n(n + 1)/2$ .

Use both identities on this page to find $21 + 23 + 25 + \cdots + 39$ without adding term by term.

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Since $21 = 2 \cdot 11 - 1$ and $39 = 2 \cdot 20 - 1$ , the sum runs from the $11$ th to the $20$ th odd number. It is the sum of the first $20$ odd numbers minus the sum of the first $10$ , so it equals

\begin{aligned} &20^2 - 10^2 \\[0.4em] &\quad = (20 - 10)(20 + 10) \\[0.4em] &\quad = 10 \times 30 = 300. \end{aligned}

In the dot picture these ten odd numbers are the outer ten layers of a $20$ by $20$ square, the L-shaped region left after removing the $10$ by $10$ square in the bottom left corner. Since $20 \ge 10 \ge 0$ , the area picture applies and rearranges that region into a $10$ by $30$ rectangle. A check by averages agrees, since the ten terms average $(21 + 39)/2 = 30$ and $10 \times 30 = 300$ .