Question 9
A group of children stand in a line. Each child says a number ('0', '1', or '2') representing the number of taller neighbors standing next to them.
Q. How would you rearrange the five children so that the maximum number of children say '2'?
To make a child say '2', both their neighbors must be taller than them.
Step 1 — Understand the '2' rule
A child says '2' only if they have two neighbors. Both neighbors must be taller than this child. In a line of five children, only the second, third, and fourth children have two neighbors. Let us call the children in the line . Only can possibly say '2'.
Step 2 — Find the maximum number of '2's
If says '2', then is taller than , and is taller than . If says '2', then is taller than , and is taller than . If both and say '2', then must be taller than (for to say '2'). Also, must be taller than (for to say '2'). This is not possible, as a child cannot be both taller and shorter than another child. So, two children standing next to each other cannot both say '2'. This means we cannot have and both say '2'. We also cannot have and both say '2'. The maximum number of children saying '2' is two. This can happen if and both say '2'. This requires a pattern where children alternate between taller and shorter. The pattern would be: Taller - Shorter - Taller - Shorter - Taller.
Step 3 — Select five children
We need to pick five children from the image. To achieve the Taller - Shorter - Taller - Shorter - Taller pattern, we need three taller children and two shorter children. Let us identify the children by their relative heights from the image. The tallest children are:
- The girl with the red hijab.
- The girl with the long braid.
- The boy in the pink striped shirt. The shortest children are:
- The girl in the red dress.
- The girl with pigtails. We will select these five children.
Step 4 — Arrange the selected children
Let us arrange the selected children in the Taller - Shorter - Taller - Shorter - Taller pattern. We place the tallest children in the 'Taller' spots and the shortest children in the 'Shorter' spots. The arrangement from left to right is:
- The girl with the red hijab (Tallest)
- The girl in the red dress (Very short)
- The girl with the long braid (Tall)
- The girl with pigtails (Short)
- The boy in the pink striped shirt (Medium-tall)
Step 5 — Verify the numbers said
Let us check what each child says in this arrangement:
- The girl with the red hijab: Her only neighbor (the girl in the red dress) is shorter. So, she says '0'.
- The girl in the red dress: Her neighbors are the girl with the red hijab and the girl with the long braid. Both are taller than her. So, she says '2'.
- The girl with the long braid: Her neighbors are the girl in the red dress and the girl with pigtails. Both are shorter than her. So, she says '0'.
- The girl with pigtails: Her neighbors are the girl with the long braid and the boy in the pink striped shirt. Both are taller than her. So, she says '2'.
- The boy in the pink striped shirt: His only neighbor (the girl with pigtails) is shorter. So, he says '0'. This arrangement results in two children saying '2', which is the maximum possible.
Answer
The five children should be arranged as follows (from left to right): (i) The girl with the red hijab (ii) The girl in the red dress (iii) The girl with the long braid (iv) The girl with pigtails (v) The boy in the pink striped shirt
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