The same children, different numbers: a number depends on what is being counted

0:00Chapter Three is called Number Play, and it opens with a picture instead of a sum. Eight children are standing in a line. Each one has a speech bubble, and in each bubble there is a number. Reading from the left: zero, two, one, one, zero, two, one, zero. That's it. No question underneath. No explanation of what the numbers mean. And that is deliberate, because the book wants you to sit with something uncomfortable first.

0:31You are looking at eight numbers, and you cannot interpret a single one of them. A number on its own tells you nothing. You have to know what it is counting. Here is the hint the book gives you, and it is one line long. Look at how tall they are. That's the whole hint. Heights. Because the moment somebody says heights, you start comparing children who are standing next to each other.

0:59And that is the move. The comparison is local. Not your height in centimetres. Just the person on my left, and the person on my right. Here is the rule, in one line. Each child says how many of the people standing immediately beside them are taller than they are. That's it. Count your neighbours who beat you on height. A child in the middle has two neighbours, so the number they call out is zero, one or two.

1:31And notice what it ignores, because it ignores nearly everything. It only cares who wins a comparison, and only against the person right beside you. Let's check the rule against the picture. I'll give the eight children heights that fit, and we'll walk down the line. The first child is at the end, so they have one neighbour, and that neighbour is shorter. Zero. Second child. A short one, with taller children on both sides. Two.

2:03Third child. Taller than the one on the left, shorter than the one on the right. One. Fourth child. The same again. One. Fifth is a tall one. Both neighbours are shorter than they are. Zero. Sixth dips down again, with taller people on either side. Two. Seventh, one. And the eighth, at the far end, is taller than its only neighbour. Zero. Zero, two, one, one, zero, two, one, zero. That is the picture, exactly.

2:37Now the page turns, and the book does the thing that makes this topic worth a video. It is the same eight children — nobody has grown, nobody has shrunk. They have swapped places. And the numbers now read: one, zero, two, zero, one, two, one, zero. Put the two rows one above the other, because that pairing is the entire argument of the section. The same children. Completely different numbers.

3:07So whatever a child's number is, it is not a property of the child. The same child says zero in one line-up and two in the next, having changed nothing but their neighbours. The number belongs to the child in that position, in that line. Now turn the whole thing around. This is where it gets properly mathematical. Instead of going from an arrangement to a row of numbers, go backwards, and ask what a row can tell us.

3:39Start with the easiest one. Can a child at either end of the line say two? No. Never. And it has nothing to do with heights. A child at the end has one neighbour. There is nobody on the other side of them. You cannot have two taller neighbours when you only have one neighbour to begin with. So if somebody hands you a row that starts with a two, you can reject it on the spot, without knowing a single height.

4:09Here is the idea that makes everything else fall out. Between every pair of children standing next to each other, draw one arrow, pointing at the shorter of the two. One arrow per gap. Eight children in a line have seven gaps, so there are seven arrows. And now look again at what each child is saying. They are counting the arrows that point at them. So all eight numbers together count all the arrows, and each arrow is counted exactly once, by the child it points at.

4:43Which means the eight numbers have to add up to seven. Not roughly. Exactly seven. Check the first row. Zero, two, one, one, zero, two, one, zero. That adds to seven. Check the second. One, zero, two, zero, one, two, one, zero. Also seven. That is not a coincidence, and it is about to do an enormous amount of work. Question. Could every child in the line say zero? Add first. Eight zeros come to zero, and we just said the total has to be seven. So something has to give.

5:24What gives is a thing I slipped past you. An arrow only exists when one of the two children is genuinely taller. If two neighbours are exactly the same height, neither one is taller, and that gap gets no arrow at all. So everybody saying zero means every gap has lost its arrow — every pair of neighbours is the same height. And if each neighbour matches the next all the way down, all eight are the same height. That is the only way.

5:56And we got there by adding up a row of numbers. Next question, and this one uses five children instead of eight, all of different heights. Could they all say one? Five children have four gaps, so four arrows. But five ones add up to five. Five is not four. Impossible. Finished — one line of arithmetic, no picture. There is a second way to see it, the one your book leads you towards.

6:26Think about the tallest child in the line. Nobody anywhere is taller, so certainly neither neighbour is. The tallest child always says zero. Always. In any line, of any length, whatever the heights. So every row contains at least one zero, and five ones has none. And notice what that kills off. Saying zero does not mean you are short — the tallest child says zero every single time. Now the big version of the question. Five children, all different heights. How many rows of numbers are actually possible?

7:07Each child says zero, one or two, and there are five of them, so on the face of it, two hundred and forty-three rows to check. Use the arrows instead. Four gaps, and each arrow points one way or the other. Two choices, four times over. Sixteen. Each of those sixteen arrow patterns gives exactly one row of numbers, and different patterns always give different rows. So out of two hundred and forty-three rows you could write down, exactly sixteen can ever happen.

7:43The other two hundred and twenty-seven are impossible, for any five children, of any heights. The row is not a free choice. It is a record of which way four arrows point, and nothing else. One last question, and it is the hardest on the page. Five children. What is the largest number of them who can say two? A two means both arrows at that child point inward. So one child saying two uses up two arrows by itself.

8:17And there are only four arrows. So at most two children can say two — and if two of them do, they have taken every arrow there is. Can that happen? Yes. Heights three, one, four, two, five. The second child is a dip, taller children both sides. Two. So is the fourth. Two. And the row comes out zero, two, zero, two, zero. In fact that is the only row of five with two twos in it.

8:47And look at that fourth child, saying two. It is not the shortest in the line. Saying two makes you a dip, not the smallest. So what was this section really about? Before the picture, the chapter asks your class where numbers do a job. A price. A house number. A bus route. A shoe size. And the point of that list is that none of those numbers means anything until you know which question it is answering.

9:19Eight children said zero, two, one, one, zero, two, one, zero, and until you knew the question, that row was noise. Once you knew it, the row turned into evidence. You could work backwards, rule arrangements out, and prove that most rows can never happen. That is the whole of number play in one page. A number is an answer, and you have to know the question. Next time we stay with neighbours, but we change what gets compared — and a number earns a status called a supercell.