Numbers can tell us things

Numbers are usually used to count. But they can also describe , they can tell us how things are arranged. Once you see this, you start spotting numbers everywhere as little stories about the world.

Concept

Imagine a row of children of different heights. Each child looks at the two neighbours standing on either side and says a number:

• $0$ if neither neighbour is taller.
• $1$ if exactly one neighbour is taller.
• $2$ if both neighbours are taller.

The two children at the ends only have one neighbour, so they say $0$ or $1$ , never $2$.

Suppose five children stand in some order and we hear the sequence "$1, 2, 0, 1, 0$" from left to right. What does that tell us? It tells us the heights are arranged in a very specific way. The third child (saying $0$) must be the tallest of the three in the middle. The second child (saying $2$) must be shorter than both neighbours. With a little detective work you can reconstruct the order of heights from the numbers alone.

This is the heart of the idea: a list of numbers can be a code for an arrangement. Different arrangements give different codes. Some codes are impossible , for example, $2$ at an end is impossible, since an end child only has one neighbour.

Once you grasp this, you can invent your own coding rules. "Say the number of girls on your left", "say how many people in front of you have a red shirt", "say how many beads to your left are smaller than yours". Each rule produces a different sequence, and each sequence captures information about the arrangement.

Numbers also tell us about quantities: the temperature today, the price of a notebook, the population of a town. Each of these numbers carries a piece of the world inside it. Mathematics is partly the art of reading those numbers carefully , not just doing sums with them.

Worked examples

Example 1. Five children stand in a line. Each says how many of their neighbours are taller than them. The sequence (from left to right) is $1, 2, 0, 1, 0$. Can such a line exist? If yes, what can you say about the third child?

• The third child says $0$, meaning neither neighbour is taller. So the third child is taller than the second and the fourth.
• The fourth says $1$ , one neighbour (either third or fifth) is taller.
• The fifth says $0$ , and being at an end, has only one neighbour (the fourth). So the fifth is taller than the fourth.
• Putting it together: child $3$ is the tallest among children $2, 3, 4$; child $5$ is taller than child $4$. Consistent , such a line can exist.

Example 2. A child says $2$. Where can this child not be standing?

• A child saying $2$ has both neighbours taller. So they need two neighbours, which means they cannot be at either end of the line.

Example 3. Three children stand in increasing order of height (shortest on the left). What sequence do they say?

• Leftmost: only the middle is a neighbour, and the middle is taller. So leftmost says $1$.
• Middle: both neighbours , left is shorter, right is taller. So middle says $1$.
• Rightmost: only the middle is a neighbour, and the middle is shorter. So rightmost says $0$.
• Sequence: $1, 1, 0$.

Example 4. Five identical children (same height) stand in a line. What sequence do they say?

• A "taller" neighbour requires a strictly taller height. If all heights are equal, no one has a taller neighbour.
• So everyone says $0$. Sequence: $0, 0, 0, 0, 0$.

Try it yourself

1. Decode the sequence $0, 2, 1, 0$ for $4$ children: what does each child's number tell you?
2. Can the sequence $2, 0, 0, 0, 2$ describe a line of $5$ children? Why or why not?
3. Five children stand in order of height , tallest on the left, shortest on the right. Write the sequence.
4. Make up your own arrangement of $6$ children and write the sequence they would say.
5. A child says $0$ at the end of the line. Is the end child shorter or taller than its only neighbour?
6. Is the sequence $1, 2, 1$ possible? If yes, describe an arrangement; if no, explain.
7. Imagine the rule changed to "say the number of shorter neighbours" instead. Repeat question $3$ with this new rule.

Activity

Line-up game. With $5$ or $6$ friends or family members of clearly different heights, line up in a random order. Each person calls out the number of taller neighbours they have. Write down the sequence. Then swap places , does the sequence change? Try to arrange yourselves so that the sequence is exactly $0, 1, 2, 1, 0$.