Question 8
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. Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
We can figure out how tall each child is compared to their neighbors.
Step 1 — Understand the rules
There are 5 children in a line. Each child tells taller neighbors. '0' means no neighbor is taller. '1' means one neighbor is taller. '2' means both neighbors are taller. End children have only one neighbor.
Step 2 — Look at the third child
The third child says 2. This child has two neighbors. Their left neighbor is the second child. Their right neighbor is the fourth child. Both neighbors must be taller. So, the second child is taller than the third. The fourth child is taller than the third. Let us write: Second child > Third child. Also: Fourth child > Third child.
Step 3 — Look at the second child
The second child says 1. This child has two neighbors. Their left neighbor is the first child. Their right neighbor is the third child. We know the second child is taller than the third. So, the third child is not taller. Only one neighbor can be taller. The first child must be taller than the second. Let us write: First child > Second child.
Step 4 — Look at the fourth child
The fourth child says 1. This child has two neighbors. Their left neighbor is the third child. Their right neighbor is the fifth child. We know the fourth child is taller than the third. So, the third child is not taller. Only one neighbor can be taller. The fifth child must be taller than the fourth. Let us write: Fifth child > Fourth child.
Step 5 — Look at the end children
The first child says 0. This child has one neighbor. That neighbor is the second child. We found the first child is taller than the second. So, the second child is not taller. This matches the first child saying 0. The fifth child says 0. This child has one neighbor. That neighbor is the fourth child. We found the fifth child is taller than the fourth. So, the fourth child is not taller. This matches the fifth child saying 0.
Step 6 — Find possible heights
All statements are consistent. We can arrange the heights. First child > Second child > Third child. Fifth child > Fourth child > Third child. Let us try some numbers. Let the third child's height be 1. The second child must be taller than 1. Let its height be 2. The fourth child must be taller than 1. Let its height be 2. The first child must be taller than 2. Let its height be 3. The fifth child must be taller than 2. Let its height be 3. Possible heights are: 3, 2, 1, 2, 3.
Step 7 — Check the heights
Let us check if these heights work. Child 1 (height 3): Neighbor Child 2 (height 2). Child 2 is not taller. Child 1 says 0. (Correct) Child 2 (height 2): Neighbors Child 1 (3), Child 3 (1). Child 1 is taller. Child 3 is not taller. Child 2 says 1. (Correct) Child 3 (height 1): Neighbors Child 2 (2), Child 4 (2). Child 2 is taller. Child 4 is taller. Child 3 says 2. (Correct) Child 4 (height 2): Neighbors Child 3 (1), Child 5 (3). Child 3 is not taller. Child 5 is taller. Child 4 says 1. (Correct) Child 5 (height 3): Neighbor Child 4 (height 2). Child 4 is not taller. Child 5 says 0. (Correct) All numbers match the sequence.
Answer
(i) The sequence 0, 1, 2, 1, 0 is possible. This is because we can find a set of heights for the children that matches what they say. For example, if the children's heights are 3, 2, 1, 2, 3 from left to right, the conditions are met.
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