Taller neighbours

Some children in a park stand in a line, and each child says a number. What do the numbers mean? Each child says how many of the children right next to them are taller.

• 2 if both neighbours are taller
• 1 if only one neighbour is taller
• 0 if neither neighbour is taller

A child in the middle of the line has two neighbours, so can say 0, 1 or 2. A child at either end has only one neighbour, so can say only 0 or 1, never 2.

Now take 5 children, all of different heights. Can they all say 1? No! Look at the tallest child: nobody is taller than them, so the tallest child always says 0. So 1, 1, 1, 1, 1 is impossible.

Is 0, 1, 2, 1, 0 possible? Yes. Put the shortest child in the middle, a taller child on each side of them, and the two tallest children at the ends. For example, heights 140, 132, 125, 130 and 138 cm, in this order, say 0, 1, 2, 1, 0.

How many of the 5 can say 2? The two end children never can. And two children standing next to each other can never both say 2, because each would have to be shorter than the other. So only the 2nd and 4th children can say 2 together: at most 2 of the 5.