Numbers that tell us things

The children from the Grade 6 textbook are back, standing in a line and calling out numbers — but by a new rule. Then they rearrange themselves and each calls out a different number. What do the numbers mean?

The rule is this: Each child calls out the number of children in front of them who are taller than they are. Look only at the people in front of you, and count only the ones taller than you; people behind you never count. Check the rule against both arrangements and see that every number fits.

Notice at once what follows. The first child in the line has nobody in front, so the first person's number must always be 0. That is not a coincidence about a particular line; it is forced by the rule.

Now work backwards. Can you build an arrangement that reads 0, 0, 0, 0, 0, 0, 0? Yes — line the children up shortest at the front, tallest at the back, so nobody ever has a taller child ahead of them.

What about 0, 1, 2, 3, 4, 5, 6? That is the reverse: tallest at the front, so each child has every earlier child taller than them.

Some statements about the rule are always true, some only sometimes, and some never. 'If a person is the tallest, then their number is 0' is ALWAYS true — nobody can be taller than the tallest. But 'if a person says 0, then they are the tallest' is only SOMETIMES true: the first person always says 0 whatever their height.

Try a few more. 'If a person is not first or last in line, they cannot say 0' — is that right? 'The person who calls out the largest number is the shortest' — must that be so? And what is the largest number possible in a group of 8 people?