Linking numbers and shapes

So far we have looked at number sequences and shape sequences separately. The real beauty appears when the two meet , when a shape sequence is literally counting out a number sequence, or when a number sequence becomes a hidden geometry. This is the deepest idea in Chapter 1.

Concept

Consider the triangular numbers again: $1, 3, 6, 10, 15, \dots$. They are not just numbers; they are pictures. The $n$th triangular number is the number of dots in a triangle of side $n$. So triangular numbers and triangles of dots are the same thing in two languages , arithmetic and geometry.

The same is true for square numbers. The $n$th square number $n^2$ is the count of dots in an $n \times n$ array. Whenever you see a $5 \times 5$ chessboard square or a window pane with $36$ panes, you are looking at a square number.

Some surprising bridges:

Sum of two consecutive triangular numbers gives a square. Try it:

• $T_1 + T_2 = 1 + 3 = 4 = 2^2$.
• $T_2 + T_3 = 3 + 6 = 9 = 3^2$.
• $T_3 + T_4 = 6 + 10 = 16 = 4^2$.

Why? Place a triangle of side $n$ next to an upside-down triangle of side $n+1$. They fit together exactly to form an $(n+1) \times (n+1)$ square. Number rule and picture proof , together.

Sum of first $n$ odd numbers equals $n^2$. We saw the gnomon proof in section 3.

Counting numbers and steps. A staircase with $n$ steps uses $1 + 2 + \dots + n = T_n$ blocks. So if you ever build a staircase out of bricks, the brick count is a triangular number.

Powers of 2 and folding paper. Fold a strip of paper in half; you get $2$ layers. Fold again; $4$ layers. Again; $8$ layers. The thickness sequence is exactly $2, 4, 8, 16, 32, \dots$ , the powers of $2$, growing as you double.

Hexagonal numbers. Pebbles arranged in concentric hexagons give the sequence $1, 7, 19, 37, \dots$. Each new ring adds $6$ more than the last ($+6, +12, +18, \dots$). Hexagonal numbers are cousins of triangulars and squares , geometry from a different shape gives a different number family.

The lesson: when a number puzzle feels hard, try drawing it. When a shape puzzle feels hard, try counting it. Mathematicians switch between the two languages constantly , it is like having two pairs of eyes.

Worked examples

Example 1. Use the rule $T_{n-1} + T_n = n^2$ to find $T_9 + T_{10}$.

• It must equal $10^2 = 100$.
• Check: $T_9 = \frac{9 \times 10}{2} = 45$ and $T_{10} = \frac{10 \times 11}{2} = 55$.
• $45 + 55 = 100$.

Example 2. A pile of bricks is laid in a staircase with $15$ steps. How many bricks?

• $T_{15} = \frac{15 \times 16}{2} = 120$.

Example 3. A square grid of dots has $144$ dots in total. How many dots on a side?

• $n^2 = 144 \Rightarrow n = 12$. So a $12 \times 12$ grid.

Example 4. Paper of thickness $0.1$ mm is folded $7$ times. How thick is the stack? (Assume folding is possible.)

• After $7$ folds, layers $= 2^7 = 128$.
• Thickness $= 128 \times 0.1$ mm $= 12.8$ mm, just over a centimetre.

Try it yourself

1. Verify that $T_5 + T_6 = 6^2$.
2. A staircase has $20$ steps. How many bricks does it use?
3. A square has $625$ small tiles arranged in a square pattern. How many tiles on a side?
4. Continue the hexagonal numbers $1, 7, 19, 37, \dots$ for two more terms.
5. Fold a paper $10$ times in your head. How many layers?
6. Find a pair of triangular numbers whose sum is $25$.
7. Challenge: Are there triangular numbers that are also square? Find one besides $1$.

Activity

Two languages, one picture. On grid paper, draw a $5 \times 5$ square of dots. Now draw a slanted line from one corner that splits the square into two triangles of dots. Count the dots in each triangle. Do they add up to $25$? Are they triangular numbers? Write the two-sentence explanation of why they had to be triangular numbers.